Source

fcla / src / section-PDM.xml

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<?xml version="1.0" encoding="UTF-8" ?>
<section acro="PDM">
<title>Properties of Determinants of Matrices</title>

<!-- %%%%%%%%%% -->
<!-- % -->
<!-- %  Section PDM -->
<!-- %  Determinants of Matrices -->
<!-- % -->
<!-- %%%%%%%%%% -->
<introduction>
<p>We have seen how to compute the determinant of a matrix, and the incredible fact that we can perform expansion about <em>any</em> row <em>or</em> column to make this computation.  In this largely theoretical section, we will state and prove several more intriguing properties about determinants.  Our main goal will be the two results in <acroref type="theorem" acro="SMZD" /> and <acroref type="theorem" acro="DRMM" />, but more specifically, we will see how the value of a determinant will allow us to gain insight into the various properties of a square matrix.</p>

</introduction>

<subsection acro="DRO">
<title>Determinants and Row Operations</title>

<p>We start easy with a straightforward theorem whose proof presages the style of subsequent proofs in this subsection.</p>

<theorem acro="DZRC" index="determinant!zero row or column">
<title>Determinant with Zero Row or Column</title>
<statement>
<p>Suppose that $A$ is a square matrix with a row where every entry is zero, or a column where every entry is zero.  Then $\detname{A}=0$.</p>

</statement>

<proof>
<p>Suppose that $A$ is a square matrix of size $n$ and row $i$ has every entry equal to zero.  We compute $\detname{A}$ via expansion about row $i$.
<alignmath>
\detname{A}
<![CDATA[&=]]>
\sum_{j=1}^{n}(-1)^{i+j}\matrixentry{A}{ij}\detname{\submatrix{A}{i}{j}}
<![CDATA[&&]]>\text{<acroref type="theorem" acro="DER" />}\\
<![CDATA[&=]]>
\sum_{j=1}^{n}(-1)^{i+j}\,0\,\detname{\submatrix{A}{i}{j}}
<![CDATA[&&\text{Row $i$ is zeros}\\]]>
<![CDATA[&=]]>
\sum_{j=1}^{n}0=0
</alignmath>
</p>

<p>The proof for the case of a zero column is entirely similar, or could be derived from an application of <acroref type="theorem" acro="DT" /> employing the transpose of the matrix.</p>

</proof>
</theorem>

<theorem acro="DRCS" index="determinant!row or column swap">
<title>Determinant for Row or Column Swap</title>
<statement>
<p>Suppose that $A$ is a square matrix.  Let $B$ be the square matrix obtained from $A$ by interchanging the location of two rows, or interchanging the location of two columns.  Then $\detname{B}=-\detname{A}$.</p>

</statement>

<proof>
<p>Begin with the special case where $A$ is a square matrix of size $n$ and we form $B$ by swapping <em>adjacent</em> rows $i$ and $i+1$ for some $1\leq i\leq n-1$.    Notice that the assumption about swapping adjacent rows means that $\submatrix{B}{i+1}{j}=\submatrix{A}{i}{j}$ for all $1\leq j\leq n$, and $\matrixentry{B}{i+1,j}=\matrixentry{A}{ij}$ for all $1\leq j\leq n$.  We compute $\detname{B}$ via expansion about row $i+1$.
<alignmath>
\detname{B}
<![CDATA[&=]]>
\sum_{j=1}^{n}(-1)^{(i+1)+j}\matrixentry{B}{i+1,j}\detname{\submatrix{B}{i+1}{j}}
<![CDATA[&&]]>\text{<acroref type="theorem" acro="DER" />}\\
<![CDATA[&=]]>
\sum_{j=1}^{n}(-1)^{(i+1)+j}\matrixentry{A}{ij}\detname{\submatrix{A}{i}{j}}
<![CDATA[&&\text{Hypothesis}\\]]>
<![CDATA[&=]]>
\sum_{j=1}^{n}(-1)^{1}(-1)^{i+j}\matrixentry{A}{ij}\detname{\submatrix{A}{i}{j}}\\
<![CDATA[&=]]>
(-1)\sum_{j=1}^{n}(-1)^{i+j}\matrixentry{A}{ij}\detname{\submatrix{A}{i}{j}}\\
<![CDATA[&=]]>
<![CDATA[-\detname{A}&&]]>\text{<acroref type="theorem" acro="DER" />}
</alignmath>
</p>

<p>So the result holds for the special case where we swap adjacent rows of the matrix.  As any computer scientist knows, we can accomplish <em>any</em> rearrangement of an ordered list by swapping adjacent elements.  This principle can be demonstrated by na\"ive sorting algorithms such as <q>bubble sort.</q>  In any event, we don't need to discuss every possible reordering, we just need to consider a swap of two rows, say rows $s$ and $t$ with
<![CDATA[$1\leq s<t\leq n$.]]>
</p>

<p>Begin with row $s$, and repeatedly swap it with each row just below it, including row $t$ and stopping there.  This will total $t-s$ swaps.  Now swap the former row $t$, which currently lives in row $t-1$, with each row above it, stopping when it becomes row $s$.  This will total another $t-s-1$ swaps.  In this way, we create $B$ through a sequence of $2(t-s)-1$ swaps of adjacent rows, each of which adjusts $\detname{A}$ by a multiplicative factor of $-1$.  So
<equation>
\detname{B}
=
(-1)^{2(t-s)-1}\detname{A}
=
\left((-1)^2\right)^{t-s}(-1)^{-1}\detname{A}
=
-\detname{A}
</equation>
as desired.</p>

<p>The proof for the case of swapping two columns is entirely similar, or could be derived from an application of <acroref type="theorem" acro="DT" /> employing the transpose of the matrix.</p>

</proof>
</theorem>

<p>So <acroref type="theorem" acro="DRCS" /> tells us the effect of the first row operation (<acroref type="definition" acro="RO" />) on the determinant of a matrix.  Here's the effect of the second row operation.</p>

<theorem acro="DRCM" index="determinant!row or column multiple">
<title>Determinant for Row or Column Multiples</title>
<statement>
<p>Suppose that $A$ is a square matrix.  Let $B$ be the square matrix obtained from $A$ by multiplying a single row by the scalar $\alpha$, or by multiplying a single column by the scalar $\alpha$.  Then $\detname{B}=\alpha\detname{A}$.</p>

</statement>

<proof>
<p>Suppose that $A$ is a square matrix of size $n$ and we form the square matrix $B$ by multiplying each entry of row $i$ of $A$ by $\alpha$.  Notice that the other rows of $A$ and $B$ are equal, so $\submatrix{A}{i}{j}=\submatrix{B}{i}{j}$, for all $1\leq j\leq n$.  We compute $\detname{B}$ via expansion about row $i$.
<alignmath>
\detname{B}
<![CDATA[&=]]>
\sum_{j=1}^{n}(-1)^{i+j}\matrixentry{B}{ij}\detname{\submatrix{B}{i}{j}}
<![CDATA[&&]]>\text{<acroref type="theorem" acro="DER" />}\\
<![CDATA[&=]]>
\sum_{j=1}^{n}(-1)^{i+j}\matrixentry{B}{ij}\detname{\submatrix{A}{i}{j}}
<![CDATA[&&\text{Hypothesis}\\]]>
<![CDATA[&=]]>
\sum_{j=1}^{n}(-1)^{i+j}\alpha\matrixentry{A}{ij}\detname{\submatrix{A}{i}{j}}
<![CDATA[&&\text{Hypothesis}\\]]>
<![CDATA[&=]]>
\alpha\sum_{j=1}^{n}(-1)^{i+j}\matrixentry{A}{ij}\detname{\submatrix{A}{i}{j}}\\
<![CDATA[&=]]>
<![CDATA[\alpha\detname{A}&&]]>\text{<acroref type="theorem" acro="DER" />}\\
</alignmath>
</p>

<p>The proof for the case of a multiple of a column is entirely similar, or could be derived from an application of <acroref type="theorem" acro="DT" /> employing the transpose of the matrix.</p>

</proof>
</theorem>

<p>Let's go for understanding the effect of all three row operations.  But first we need an intermediate result, but it is an easy one.</p>

<theorem acro="DERC" index="determinant!equal rows or columns">
<title>Determinant with Equal Rows or Columns</title>
<statement>
<p>Suppose that $A$ is a square matrix with two equal rows, or two equal columns.  Then $\detname{A}=0$.</p>

</statement>

<proof>
<p>Suppose that $A$ is a square matrix of size $n$ where the two rows $s$ and $t$ are equal.  Form the matrix $B$ by swapping rows $s$ and $t$.  Notice that as a consequence of our hypothesis, $A=B$.  Then
<alignmath>
\detname{A}
<![CDATA[&=]]>
\frac{1}{2}\left(\detname{A}+\detname{A}\right)\\
<![CDATA[&=]]>
\frac{1}{2}\left(\detname{A}-\detname{B}\right)
<![CDATA[&&]]>\text{<acroref type="theorem" acro="DRCS" />}\\
<![CDATA[&=]]>
\frac{1}{2}\left(\detname{A}-\detname{A}\right)
<![CDATA[&&\text{Hypothesis, $A=B$}\\]]>
<![CDATA[&=]]>
\frac{1}{2}\,(0)=0
</alignmath>
</p>

<p>The proof for the case of two equal columns is entirely similar, or could be derived from an application of <acroref type="theorem" acro="DT" /> employing the transpose of the matrix.</p>

</proof>
</theorem>

<p>Now explain the third row operation.  Here we go.</p>

<theorem acro="DRCMA" index="indexstring">
<title>Determinant for Row or Column Multiples and Addition</title>
<statement>
<p>Suppose that $A$ is a square matrix.  Let $B$ be the square matrix obtained from $A$ by multiplying a row by the scalar $\alpha$ and then adding it to another row, or by multiplying a column by the scalar $\alpha$ and then adding it to another column.  Then $\detname{B}=\detname{A}$.</p>

</statement>

<proof>
<p>Suppose that $A$ is a square matrix of size $n$.  Form the matrix $B$ by multiplying row $s$ by $\alpha$ and adding it to row $t$.  Let $C$ be the auxiliary matrix where we replace row $t$ of $A$ by row $s$ of $A$.  Notice that $\submatrix{A}{t}{j}=\submatrix{B}{t}{j}=\submatrix{C}{t}{j}$ for all $1\leq j\leq n$.  We compute the determinant of $B$ by expansion about row $t$.
<alignmath>
\detname{B}
<![CDATA[&=]]>
\sum_{j=1}^{n}(-1)^{t+j}\matrixentry{B}{tj}\detname{\submatrix{B}{t}{j}}
<![CDATA[&&]]>\text{<acroref type="theorem" acro="DER" />}\\
<![CDATA[&=]]>
\sum_{j=1}^{n}(-1)^{t+j}\left(\alpha\matrixentry{A}{sj}+\matrixentry{A}{tj}\right)\detname{\submatrix{B}{t}{j}}
<![CDATA[&&\text{Hypothesis}\\]]>
<![CDATA[&=]]>
\sum_{j=1}^{n}(-1)^{t+j}\alpha\matrixentry{A}{sj}\detname{\submatrix{B}{t}{j}}\\
<![CDATA[&\quad\quad+]]>
\sum_{j=1}^{n}(-1)^{t+j}\matrixentry{A}{tj}\detname{\submatrix{B}{t}{j}}\\
<![CDATA[&=]]>
\alpha\sum_{j=1}^{n}(-1)^{t+j}\matrixentry{A}{sj}\detname{\submatrix{B}{t}{j}}\\
<![CDATA[&\quad\quad+]]>
\sum_{j=1}^{n}(-1)^{t+j}\matrixentry{A}{tj}\detname{\submatrix{B}{t}{j}}\\
<![CDATA[&=]]>
\alpha\sum_{j=1}^{n}(-1)^{t+j}\matrixentry{C}{tj}\detname{\submatrix{C}{t}{j}}\\
<![CDATA[&\quad\quad+]]>
\sum_{j=1}^{n}(-1)^{t+j}\matrixentry{A}{tj}\detname{\submatrix{A}{t}{j}}\\
<![CDATA[&=]]>
<![CDATA[\alpha\detname{C}+\detname{A}&&]]>\text{<acroref type="theorem" acro="DER" />}\\
<![CDATA[&=]]>
<![CDATA[\alpha\,0+\detname{A} = \detname{A}&&]]>\text{<acroref type="theorem" acro="DERC" />}
</alignmath>
</p>

<p>The proof for the case of adding a multiple of a column is entirely similar, or could be derived from an application of <acroref type="theorem" acro="DT" /> employing the transpose of the matrix.</p>

</proof>
</theorem>

<p>Is this what you expected?  We could argue that the third row operation is the most popular, and yet it has no effect whatsoever on the determinant of a matrix!  We can exploit this, along with our understanding of the other two row operations, to provide another approach to computing a determinant.  We'll explain this in the context of an example.</p>

<example acro="DRO" index="determinant!via row operations">
<title>Determinant by row operations</title>

<p>Suppose we desire the determinant of the $4\times 4$ matrix
<equation>
A=
\begin{bmatrix}
<![CDATA[2 & 0 & 2 & 3 \\]]>
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[-1& 1 &-1 & 2 \\]]>
<![CDATA[3 & 5 & 4 & 0]]>
\end{bmatrix}
</equation>
</p>

<p>We will perform a sequence of row operations on this matrix, shooting for an upper triangular matrix, whose determinant will be simply the product of its diagonal entries.  For each row operation, we will track the effect on the determinant via <acroref type="theorem" acro="DRCS" />, <acroref type="theorem" acro="DRCM" />, <acroref type="theorem" acro="DRCMA" />.
<alignmath>
<![CDATA[A&=]]>
\begin{bmatrix}
<![CDATA[2 & 0 & 2 & 3 \\]]>
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[-1& 1 &-1 & 2 \\]]>
<![CDATA[3 & 5 & 4 & 0]]>
\end{bmatrix}    
<![CDATA[&&\detname{A}&&]]>\\
\xrightarrow{\rowopswap{1}{2}}
<![CDATA[A_1&=]]>
\begin{bmatrix}
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[2 & 0 & 2 & 3 \\]]>
<![CDATA[-1& 1 &-1 & 2 \\]]>
<![CDATA[3 & 5 & 4 & 0]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[&=-\detname{A_1}&&]]>\text{<acroref type="theorem" acro="DRCS" />}\\
\xrightarrow{\rowopadd{-2}{1}{2}}
<![CDATA[A_2&=]]>
\begin{bmatrix}
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[0 &-6 & 4 & 1 \\]]>
<![CDATA[-1& 1 &-1 & 2 \\]]>
<![CDATA[3 & 5 & 4 & 0]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[&=-\detname{A_2}&&]]>\text{<acroref type="theorem" acro="DRCMA" />}\\
\xrightarrow{\rowopadd{1}{1}{3}}
<![CDATA[A_3&=]]>
\begin{bmatrix}
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[0 &-6 & 4 & 1 \\]]>
<![CDATA[0 & 4 &-2 & 3 \\]]>
<![CDATA[3 & 5 & 4 & 0]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[&=-\detname{A_3}&&]]>\text{<acroref type="theorem" acro="DRCMA" />}\\
\xrightarrow{\rowopadd{-3}{1}{4}}
<![CDATA[A_4&=]]>
\begin{bmatrix}
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[0 &-6 & 4 & 1 \\]]>
<![CDATA[0 & 4 &-2 & 3 \\]]>
<![CDATA[0 &-4 & 7 &-3]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[&=-\detname{A_4}&&]]>\text{<acroref type="theorem" acro="DRCMA" />}\\
\xrightarrow{\rowopadd{1}{3}{2}}
<![CDATA[A_5&=]]>
\begin{bmatrix}
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[0 &-2 & 2 & 4 \\]]>
<![CDATA[0 & 4 &-2 & 3 \\]]>
<![CDATA[0 &-4 & 7 &-3]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[&=-\detname{A_5}&&]]>\text{<acroref type="theorem" acro="DRCMA" />}\\
\xrightarrow{\rowopmult{-\frac{1}{2}}{2}}
<![CDATA[A_6&=]]>
\begin{bmatrix}
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[0 & 1 &-1 &-2 \\]]>
<![CDATA[0 & 4 &-2 & 3 \\]]>
<![CDATA[0 &-4 & 7 &-3]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[&=2\detname{A_6}&&]]>\text{<acroref type="theorem" acro="DRCM" />}\\
\xrightarrow{\rowopadd{-4}{2}{3}}
<![CDATA[A_7&=]]>
\begin{bmatrix}
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[0 & 1 &-1 &-2 \\]]>
<![CDATA[0 & 0 & 2 &11 \\]]>
<![CDATA[0 &-4 & 7 &-3]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[&=2\detname{A_7}&&]]>\text{<acroref type="theorem" acro="DRCMA" />}\\
\xrightarrow{\rowopadd{4}{2}{4}}
<![CDATA[A_8&=]]>
\begin{bmatrix}
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[0 & 1 &-1 &-2 \\]]>
<![CDATA[0 & 0 & 2 &11 \\]]>
<![CDATA[0 & 0 & 3 &-11]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[&=2\detname{A_8}&&]]>\text{<acroref type="theorem" acro="DRCMA" />}\\
\xrightarrow{\rowopadd{-1}{3}{4}}
<![CDATA[A_9&=]]>
\begin{bmatrix}
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[0 & 1 &-1 &-2 \\]]>
<![CDATA[0 & 0 & 2 &11 \\]]>
<![CDATA[0 & 0 & 1 &-22]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[&=2\detname{A_9}&&]]>\text{<acroref type="theorem" acro="DRCMA" />}\\
\xrightarrow{\rowopadd{-2}{4}{3}}
<![CDATA[A_{10}&=]]>
\begin{bmatrix}
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[0 & 1 &-1 &-2 \\]]>
<![CDATA[0 & 0 & 0 &55 \\]]>
<![CDATA[0 & 0 & 1 &-22]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[&=2\detname{A_{10}}&&]]>\text{<acroref type="theorem" acro="DRCMA" />}\\
\xrightarrow{\rowopswap{3}{4}}
<![CDATA[A_{11}&=]]>
\begin{bmatrix}
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[0 & 1 &-1 &-2 \\]]>
<![CDATA[0 & 0 & 1 &-22\\]]>
<![CDATA[0 & 0 & 0 &55]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[&=-2\detname{A_{11}}&&]]>\text{<acroref type="theorem" acro="DRCS" />}\\
\xrightarrow{\rowopmult{\frac{1}{55}}{4}}
<![CDATA[A_{12}&=]]>
\begin{bmatrix}
<![CDATA[1 & 3 &-1 & 1 \\]]>
<![CDATA[0 & 1 &-1 &-2 \\]]>
<![CDATA[0 & 0 & 1 &-22\\]]>
<![CDATA[0 & 0 & 0 & 1]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[&=-110\detname{A_{12}}&&]]>\text{<acroref type="theorem" acro="DRCM" />}\\
</alignmath>
</p>

<p>The matrix $A_{12}$ is upper triangular, so expansion about the first column (repeatedly) will result in $\detname{A_{12}}=(1)(1)(1)(1)=1$ (see <acroref type="example" acro="DUTM" />) and thus, $\detname{A}=-110(1)=-110$.</p>

<p>Notice that our sequence of row operations was somewhat <i>ad hoc</i>, such as the transformation to $A_5$.  We could have been even more methodical, and strictly followed the process that converts a matrix to reduced row-echelon form (<acroref type="theorem" acro="REMEF" />), eventually achieving the same numerical result with a final matrix that equaled the $4\times 4$ identity matrix.  Notice too that we could have stopped with $A_8$, since at this point we could compute $\detname{A_8}$ by two expansions about first columns, followed by a simple determinant of a $2\times 2$ matrix (<acroref type="theorem" acro="DMST" />).</p>

<p>The beauty of this approach is that computationally we should already have written a procedure to convert matrices to reduced-row echelon form, so all we need to do is track the multiplicative changes to the determinant as the algorithm proceeds.   Further, for a square matrix of size $n$ this approach requires on the order of $n^3$ multiplications, while a recursive application of expansion about a row or column (<acroref type="theorem" acro="DER" />, <acroref type="theorem" acro="DEC" />) will require in the vicinity of $(n-1)(n!)$ multiplications.  So even for very small matrices, a computational approach utilizing row operations will have superior run-time.  Tracking, and controlling, the effects of round-off errors is another story, best saved for a numerical linear algebra course.</p>

</example>

</subsection>

<subsection acro="DROEM">
<title>Determinants, Row Operations, Elementary Matrices</title>

<p>As a final preparation for our two most important theorems about determinants, we prove a handful of facts about the interplay of row operations and matrix multiplication with elementary matrices with regard to the determinant.  But first, a simple, but crucial, fact about the identity matrix.</p>

<theorem acro="DIM" index="determinant!identity matrix">
<title>Determinant of the Identity Matrix</title>
<statement>
<indexlocation index="identity matrix!determinant" />
<p>For every $n\geq 1$, $\detname{I_n}=1$.</p>

</statement>

<proof>
<p>It may be overkill, but this is a good situation to run through a proof by induction on $n$ (<acroref type="technique" acro="I" />).  Is the result true when $n=1$? Yes,
<alignmath>
\detname{I_1}
<![CDATA[&=\matrixentry{I_1}{11}&&]]>\text{<acroref type="definition" acro="DM" />}\\
<![CDATA[&=1&&]]>\text{<acroref type="definition" acro="IM" />}\\
</alignmath>
</p>

<p>Now assume the theorem is true for the identity matrix of size $n-1$ and investigate the determinant of the identity matrix of size $n$ with expansion about row 1,
<alignmath>
\detname{I_n}
<![CDATA[&=]]>
\sum_{j=1}^{n}(-1)^{1+j}\matrixentry{I_n}{1j}\detname{\submatrix{I_n}{1}{j}}
<![CDATA[&&]]>\text{<acroref type="definition" acro="DM" />}\\
<![CDATA[&=]]>
(-1)^{1+1}\matrixentry{I_n}{11}\detname{\submatrix{I_n}{1}{1}}\\
<![CDATA[&\quad\quad+]]>
\sum_{j=2}^{n}(-1)^{1+j}\matrixentry{I_n}{1j}\detname{\submatrix{I_n}{1}{j}}\\
<![CDATA[&=]]>
1\detname{I_{n-1}}+
\sum_{j=2}^{n}(-1)^{1+j}\,0\,\detname{\submatrix{I_n}{1}{j}}
<![CDATA[&&]]>\text{<acroref type="definition" acro="IM" />}\\
<![CDATA[&=]]>
1(1)+\sum_{j=2}^{n}\,0=1
<![CDATA[&&\text{Induction Hypothesis}\\]]>
</alignmath>
</p>

</proof>
</theorem>

<theorem acro="DEM" index="elementary matrices!determinants">
<title>Determinants of Elementary Matrices</title>
<statement>
<p>For the three possible versions of an elementary matrix (<acroref type="definition" acro="ELEM" />) we have the determinants,
<ol><li> $\detname{\elemswap{i}{j}}=-1$
</li><li> $\detname{\elemmult{\alpha}{i}}=\alpha$
</li><li> $\detname{\elemadd{\alpha}{i}{j}}=1$
</li></ol>
</p>
</statement>

<proof>
<p>Swapping rows $i$ and $j$ of the identity matrix will create $\elemswap{i}{j}$  (<acroref type="definition" acro="ELEM" />), so
<alignmath>
\detname{\elemswap{i}{j}}
<![CDATA[&=-\detname{I_n}&&]]>\text{<acroref type="theorem" acro="DRCS" />}\\
<![CDATA[&=-1&&]]>\text{<acroref type="theorem" acro="DIM" />}\\
</alignmath>
</p>

<!-- % -->
<p>Multiplying row $i$ of the identity matrix by $\alpha$ will create $\elemmult{\alpha}{i}$ (<acroref type="definition" acro="ELEM" />), so
<alignmath>
\detname{\elemmult{\alpha}{i}}
<![CDATA[&=\alpha\detname{I_n}&&]]>\text{<acroref type="theorem" acro="DRCM" />}\\
<![CDATA[&=\alpha(1)=\alpha&&]]>\text{<acroref type="theorem" acro="DIM" />}\\
</alignmath>
</p>

<!-- % -->
<p>Multiplying row $i$ of the identity matrix by $\alpha$ and adding to row $j$ will create $\elemadd{\alpha}{i}{j}$ (<acroref type="definition" acro="ELEM" />), so
<alignmath>
\detname{\elemadd{\alpha}{i}{j}}
<![CDATA[&=\detname{I_n}&&]]>\text{<acroref type="theorem" acro="DRCMA" />}\\
<![CDATA[&=1&&]]>\text{<acroref type="theorem" acro="DIM" />}\\
</alignmath>
</p>

<!-- % -->
</proof>
</theorem>

<theorem acro="DEMMM" index="determinants!elementary matrices">
<title>Determinants, Elementary Matrices, Matrix Multiplication</title>
<statement>
<p>Suppose that $A$ is a square matrix of size $n$ and $E$ is any elementary matrix of size $n$.  Then
<equation>
\detname{EA}=\detname{E}\detname{A}
</equation>
</p>

</statement>

<proof>
<p>The proof procedes in three parts, one for each type of elementary matrix, with each part very similar to the other two.</p>

<p>First, let $B$ be the matrix obtained from $A$ by swapping rows $i$ and $j$,
<alignmath>
\detname{\elemswap{i}{j}A}
<![CDATA[&=\detname{B}&&]]>\text{<acroref type="theorem" acro="EMDRO" />}\\
<![CDATA[&=-\detname{A}&&]]>\text{<acroref type="theorem" acro="DRCS" />}\\
<![CDATA[&=\detname{\elemswap{i}{j}}\detname{A}&&]]>\text{<acroref type="theorem" acro="DEM" />}
</alignmath>
</p>

<p>Second, let $B$ be the matrix obtained from $A$ by multiplying row $i$ by $\alpha$,
<alignmath>
\detname{\elemmult{\alpha}{i}A}
<![CDATA[&=\detname{B}&&]]>\text{<acroref type="theorem" acro="EMDRO" />}\\
<![CDATA[&=\alpha\detname{A}&&]]>\text{<acroref type="theorem" acro="DRCM" />}\\
<![CDATA[&=\detname{\elemmult{\alpha}{i}}\detname{A}&&]]>\text{<acroref type="theorem" acro="DEM" />}
</alignmath>
</p>

<p>Third, let $B$ be the matrix obtained from $A$ by multiplying row $i$ by $\alpha$ and adding to row $j$,
<alignmath>
\detname{\elemadd{\alpha}{i}{j}A}
<![CDATA[&=\detname{B}&&]]>\text{<acroref type="theorem" acro="EMDRO" />}\\
<![CDATA[&=\detname{A}&&]]>\text{<acroref type="theorem" acro="DRCMA" />}\\
<![CDATA[&=\detname{\elemadd{\alpha}{i}{j}}\detname{A}&&]]>\text{<acroref type="theorem" acro="DEM" />}
</alignmath>
</p>

<p>Since the desired result holds for each variety of elementary matrix individually, we are done.
</p>

</proof>
</theorem>

</subsection>

<subsection acro="DNMMM">
<title>Determinants, Nonsingular Matrices, Matrix Multiplication</title>

<p>If you asked someone with substantial experience working with matrices about the value of the determinant, they'd be likely to quote the following theorem as the first thing to come to mind.</p>

<theorem acro="SMZD" index="determinant!zero">
<title>Singular Matrices have Zero Determinants</title>
<statement>
<indexlocation index="determinant!nonsingular matrix" />
<p>Let $A$ be a square matrix.  Then $A$ is singular if and only if $\detname{A}=0$.</p>

</statement>

<proof>
<p>Rather than jumping into the two halves of the equivalence, we first establish a few items.  Let $B$ be the unique square matrix that is row-equivalent to $A$ and in reduced row-echelon form (<acroref type="theorem" acro="REMEF" />, <acroref type="theorem" acro="RREFU" />).  For each of the row operations that converts $B$ into $A$, there is an elementary matrix $E_i$ which effects the row operation by matrix multiplication (<acroref type="theorem" acro="EMDRO" />).  Repeated applications of <acroref type="theorem" acro="EMDRO" /> allow us to write
<equation>
A=E_sE_{s-1}\dots E_2E_1B
</equation>
</p>

<p>Then
<alignmath>
\detname{A}
<![CDATA[&=]]>
\detname{E_sE_{s-1}\dots E_2E_1B}\\
<![CDATA[&=\detname{E_s}\detname{E_{s-1}}\dots\detname{E_2}\detname{E_1}\detname{B}]]>
<![CDATA[&&]]>\text{<acroref type="theorem" acro="DEMMM" />}
</alignmath>
</p>

<p>From <acroref type="theorem" acro="DEM" /> we can infer that the determinant of an elementary matrix is never zero (note the ban on $\alpha=0$ for $\elemmult{\alpha}{i}$ in <acroref type="definition" acro="ELEM" />).  So the product on the right is composed of nonzero scalars, with the possible exception of $\detname{B}$.  More precisely, we can argue that $\detname{A}=0$ if and only if $\detname{B}=0$.  With this established, we can take up the two halves of the equivalence.</p>

<p><implyforward /> If $A$ is singular, then by <acroref type="theorem" acro="NMRRI" />, $B$ cannot be the identity matrix.  Because (1) the number of pivot columns is equal to the number of nonzero rows, (2) not every column is a pivot column, and (3) $B$ is square, we see that $B$ must have a zero row.  By <acroref type="theorem" acro="DZRC" /> the determinant of $B$ is zero, and by the above, we conclude that the determinant of $A$ is zero.</p>

<p><implyreverse /> We will prove the contrapositive (<acroref type="technique" acro="CP" />).  So assume $A$ is nonsingular, then by  <acroref type="theorem" acro="NMRRI" />, $B$ is the identity matrix and <acroref type="theorem" acro="DIM" /> tells us that $\detname{B}=1\neq 0$.  With the argument above, we conclude that the determinant of $A$ is nonzero as well.</p>

</proof>
</theorem>

<p>For the case of $2\times 2$ matrices you might compare the application of <acroref type="theorem" acro="SMZD" /> with the combination of the results stated in <acroref type="theorem" acro="DMST" /> and <acroref type="theorem" acro="TTMI" />.</p>

<example acro="ZNDAB" index="determinant!zero versus nonzero">
<title>Zero and nonzero determinant, Archetypes A and B</title>

<p>The coefficient matrix in <acroref type="archetype" acro="A" /> has a zero determinant (check this!) while the coefficient matrix <acroref type="archetype" acro="B" /> has a nonzero determinant (check this, too).  These matrices are singular and nonsingular, respectively.  This is exactly what <acroref type="theorem" acro="SMZD" /> says, and continues our list of contrasts between these two archetypes.</p>

</example>

<p>Since <acroref type="theorem" acro="SMZD" /> is an equivalence (<acroref type="technique" acro="E" />) we can expand on our growing list of equivalences about nonsingular matrices.  The addition of the condition $\detname{A}\neq 0$ is one of the best motivations for learning about determinants.</p>

<theorem acro="NME7" index="nonsingular matrix!equivalences">
<title>Nonsingular Matrix Equivalences, Round 7</title>
<statement>
<p>Suppose that $A$ is a square matrix of size $n$.  The following are equivalent.
<ol> <li>$A$ is nonsingular.
</li><li>$A$ row-reduces to the identity matrix.
</li><li>The null space of $A$ contains only the zero vector, $\nsp{A}=\set{\zerovector}$.
</li><li>The linear system $\linearsystem{A}{\vect{b}}$ has a unique solution for every possible choice of $\vect{b}$.
</li><li>The columns of $A$ are a linearly independent set.
</li><li>$A$ is invertible.
</li><li>The column space of $A$ is $\complex{n}$, $\csp{A}=\complex{n}$.
</li><li>The columns of $A$ are a basis for $\complex{n}$.
</li><li>The rank of $A$ is $n$, $\rank{A}=n$.
</li><li>The nullity of $A$ is zero, $\nullity{A}=0$.
</li><li>The determinant of $A$ is nonzero, $\detname{A}\neq 0$.
</li></ol>
</p>

</statement>

<proof>
<p><acroref type="theorem" acro="SMZD" /> says $A$ is singular if and only if $\detname{A}=0$.  If we negate each of these statements, we arrive at two contrapositives that we can combine as the equivalence, $A$ is nonsingular if and only if $\detname{A}\neq 0$.  This allows us to add a new statement to the list found in <acroref type="theorem" acro="NME6" />.</p>

</proof>
</theorem>

<p>Computationally, row-reducing a matrix is the most efficient way to  determine if a matrix is nonsingular, though the effect of using division in a computer can lead to round-off errors that confuse small quantities with critical zero quantities.  Conceptually, the determinant may seem the most efficient way to determine if a matrix is nonsingular.  The definition of a determinant uses just addition, subtraction and multiplication, so division is never a problem.  And the final test is easy:  is the determinant zero or not?  However, the number of operations involved in computing a determinant by the definition very quickly becomes so excessive as to be impractical.</p>

<p>Now for the <i>coup de grace</i>.  We will generalize <acroref type="theorem" acro="DEMMM" /> to the case of <em>any</em> two square matrices.   You may recall thinking that matrix multiplication was defined in a needlessly complicated manner.  For sure, the definition of a determinant seems even stranger.  (Though <acroref type="theorem" acro="SMZD" /> might be forcing you to reconsider.)  Read the statement of the next theorem and contemplate how nicely matrix multiplication and determinants play with each other.</p>

<theorem acro="DRMM" index="determinant!matrix multiplication">
<title>Determinant Respects Matrix Multiplication</title>
<statement>
<p>Suppose that $A$ and $B$ are square matrices of the same size.  Then $\detname{AB}=\detname{A}\detname{B}$.</p>

</statement>

<proof>
<p>This proof is constructed in two cases.  First, suppose that $A$ is singular.  Then $\detname{A}=0$ by <acroref type="theorem" acro="SMZD" />.  By the contrapositive of <acroref type="theorem" acro="NPNT" />, $AB$ is singular as well.
So by a second application of<acroref type="theorem" acro="SMZD" />, $\detname{AB}=0$.  Putting it all together
<alignmath>
\detname{AB}=0=0\,\detname{B}=\detname{A}\detname{B}
</alignmath>
as desired.</p>

<p>For the second case, suppose that $A$ is nonsingular.  By <acroref type="theorem" acro="NMPEM" /> there are elementary matrices $E_{1},\,E_{2},\,E_{3},\,\dots,\,E_{s}$ such that $A=E_1E_2E_3\dots E_s$.
Then
<alignmath>
\detname{AB}
<![CDATA[&=]]>
\detname{E_1E_2E_3\dots E_{s}B}\\
<![CDATA[&=]]>
\detname{E_1}\detname{E_2}\detname{E_3}\dots\detname{E_s}\detname{B}
<![CDATA[&&]]>\text{<acroref type="theorem" acro="DEMMM" />}\\
<![CDATA[&=]]>
\detname{E_1E_2E_3\dots E_s}\detname{B}
<![CDATA[&&]]>\text{<acroref type="theorem" acro="DEMMM" />}\\
<![CDATA[&=]]>
\detname{A}\detname{B}
</alignmath>
</p>

</proof>
</theorem>

<p>It is amazing that matrix multiplication and the determinant interact this way.  Might it also be true that $\detname{A+B}=\detname{A}+\detname{B}$?  (<acroref type="exercise" acro="PDM.M30" />)</p>

<sageadvice acro="NME7" index="nonsingular matrices!round 7">
<title>Nonsingular Matrices, Round 7</title>
Our latest addition to the NME series is sometimes considered the best computational way to determine if a matrix is nonsingular.  While computing a single number for a comparison to zero is a very appealing process, other properties of nonsingular matrices are faster for determining if a matrix is nonsingular.  Still, it is a worthwhile equivalence.  We illustrate both directions.
<sage>
<input>A = matrix(QQ, [[ 1,  0,  2,  1,  1, -3,  4, -6],
                [ 0,  1,  0,  5,  3, -8,  0,  6],
                [ 1,  1,  4,  4,  2, -8,  4, -2],
                [ 0,  1,  6,  0, -2, -1,  1,  0],
                [-1,  1,  0,  1,  0, -2, -1,  5],
                [ 0, -1, -6, -3,  1,  4,  1, -5],
                [-1, -1, -2, -3, -2,  7, -4,  2],
                [ 0,  2,  0,  0, -2, -2,  0,  6]])
A.is_singular()
</input>
<output>False
</output>
</sage>

<sage>
<input>A.determinant()
</input>
<output>4
</output>
</sage>

<sage>
<input>B = matrix(QQ, [[ 2, -1,  0,  4, -6, -2,  8, -2],
                [-1,  1, -1, -4, -1,  7, -7, -2],
                [-1,  1,  1,  0,  7, -5, -6,  7],
                [ 2, -1, -1,  3, -5, -2,  5,  2],
                [ 1, -2,  0,  2, -1, -3,  8,  0],
                [-1,  1,  0, -2,  6, -1, -8,  5],
                [ 2, -1, -1,  2, -7,  1,  7, -3],
                [-1,  0,  1,  0,  6, -4, -2,  4]])
B.is_singular()
</input>
<output>True
</output>
</sage>

<sage>
<input>B.determinant()
</input>
<output>0
</output>
</sage>



</sageadvice>
<sageadvice acro="PDM" index="properties of determinants">
<title>Properties of Determinants of Matrices</title>
There are many properties of determinants detailed in this section.  You have the tools in Sage to explore all of them.  We will illustrate with just two.  One is not particularly deep, but we personally find the interplay between matrix multiplication and the determinant nothing short of amazing, so we will not pass up the chance to verify <acroref type="theorem" acro="DRMM" />.  Add some checks of other properties yourself.<br /><br />
We copy the sixth row of the matrix <code>C</code> to the fourth row, so by <acroref type="theorem" acro="DERC" />, the determinant is zero.
<sage>
<input>C = matrix(QQ, [[-3,  4,  0,  1,  2,  0,  5,  0],
                [ 4,  0, -4,  7,  0,  5,  0,  5],
                [ 7,  4,  2,  2,  4,  0, -2,  8],
                [ 5, -4,  8,  2,  6, -1, -4, -4],
                [ 8,  0,  7,  4,  7,  5,  2, -3],
                [ 6,  5,  3,  7,  4,  2,  4, -3],
                [ 1,  6, -4, -4,  3,  8,  5, -2],
                [ 2,  4, -2,  8,  2,  5,  2,  2]])
C[3] = C[5]
C
</input>
<output>[-3  4  0  1  2  0  5  0]
[ 4  0 -4  7  0  5  0  5]
[ 7  4  2  2  4  0 -2  8]
[ 6  5  3  7  4  2  4 -3]
[ 8  0  7  4  7  5  2 -3]
[ 6  5  3  7  4  2  4 -3]
[ 1  6 -4 -4  3  8  5 -2]
[ 2  4 -2  8  2  5  2  2]
</output>
</sage>

<sage>
<input>C.determinant()
</input>
<output>0
</output>
</sage>

Now, a demonstration of <acroref type="theorem" acro="DRMM" />.
<sage>
<input>A = matrix(QQ, [[-4,  7, -2,  6,  8,  0, -4,  6],
                [ 1,  6,  5,  8,  2,  3,  1, -1],
                [ 5,  0,  7, -3,  7, -3,  6, -3],
                [-4,  5,  8,  3,  6,  8, -1, -1],
                [ 0,  0, -3, -3,  4,  4,  2,  5],
                [-3, -2, -3,  8,  8, -3, -1,  1],
                [-4,  0,  2,  4,  4,  4,  7,  2],
                [ 3,  3, -4,  5, -2,  3, -1,  5]])
B = matrix(QQ, [[ 1,  2, -4,  8, -1,  2, -1, -3],
                [ 3,  3, -2, -4, -3,  8,  1,  6],
                [ 1,  8,  4,  0,  4, -2,  0,  8],
                [ 6,  8,  1, -1, -4, -3, -2,  5],
                [ 0,  5,  1,  4, -3,  2, -3, -2],
                [ 2,  4,  0,  7,  8, -1,  5,  8],
                [ 7,  1,  1, -1, -1,  7, -2,  6],
                [ 2,  3,  4,  7,  3,  4,  7, -2]])
C = A*B; C
</input>
<output>[ 35  99  28  12 -51  46  25  16]
[ 83 144  11  -3 -17  20 -11 141]
[ 24  62   6  23 -25  52 -68  30]
[ 44 153  42  19  53   0  20 169]
[ 11   5  11  80  33  53  45 -13]
[ 25  58  23  -5 -79 -24 -45 -35]
[ 83  89  47  15  15  37   4 110]
[ 47  39 -12  56  -2  29  48  14]
</output>
</sage>

<sage>
<input>Adet = A.determinant(); Adet
</input>
<output>9350730
</output>
</sage>

<sage>
<input>Bdet = B.determinant(); Bdet
</input>
<output>6516260
</output>
</sage>

<sage>
<input>Cdet = C.determinant(); Cdet
</input>
<output>60931787869800
</output>
</sage>

<sage>
<input>Adet*Bdet == Cdet
</input>
<output>True
</output>
</sage>

Earlier, in <acroref type="sage" acro="NME1" /> we used the <code>random_matrix()</code> constructor with the <code>algorithm='unimodular'</code> keyword to create a nonsingular matrix.  We can now reveal that in this context, <q>unimodular</q> means <q>with determinant 1.</q>  So such a matrix will always be nonsingular by <acroref type="theorem" acro="SMZD" />.  But more can be said.  It is not obvious at all, and <acroref type="solution" acro="PDM.SOL.M20" /> has just a partial explanation, but the inverse of a unimodular matrix with all integer entries will have an inverse with all integer entries.<br /><br />
With a fraction-free inverse many <q>textbook</q> exercises can be constructed through the use of unimodular matrices.  Experiment for yourself below.  An <code>upper_bound</code> keyword can be used to control the size of the entries of the matrix, though this will have little control over the inverse.
<sage>
<input>A = random_matrix(QQ, 5, algorithm='unimodular')
A, A.inverse()                          # random
</input>
<output>(
[  -9  -32 -118  273   78]  [-186   30   22  -18  375]
[   2    9   31  -69  -21]  [  52   -8   -8    3 -105]
[   4   15   54 -120  -39]  [-147   25   19  -12  297]
[  -3  -11  -38   84   28]  [ -47    8    6   -4   95]
[  -5  -18  -66  152   44], [ -58   10    7   -5  117]
)
</output>
</sage>



</sageadvice>
</subsection>

<!--   End of  pdm.tex -->
<readingquestions>
<ol>
<li>Consider the two matrices below, and suppose you already have computed $\detname{A}=-120$.  What is $\detname{B}$?  Why?
<alignmath>
<![CDATA[A&=]]>
\begin{bmatrix}
<![CDATA[ 0 & 8 & 3 & -4 \\]]>
<![CDATA[ -1 & 2 & -2 & 5 \\]]>
<![CDATA[ -2 & 8 & 4 & 3 \\]]>
<![CDATA[ 0 & -4 & 2 & -3]]>
\end{bmatrix}
<![CDATA[&]]>
<![CDATA[B&=]]>
\begin{bmatrix}
<![CDATA[ 0 & 8 & 3 & -4 \\]]>
<![CDATA[ 0 & -4 & 2 & -3 \\]]>
<![CDATA[ -2 & 8 & 4 & 3 \\]]>
<![CDATA[ -1 & 2 & -2 & 5]]>
\end{bmatrix}
</alignmath>
</li>
<li>State the theorem that allows us to make yet another extension to our NMEx series of theorems.
</li>
<li>What is amazing about the interaction between matrix multiplication and the determinant?
</li></ol>
</readingquestions>

<exercisesubsection>

<exercise type="C" number="30" rough="determinants of square archetypes">
<problem contributor="robertbeezer">Each of the archetypes below is a system of equations with a square coefficient matrix, or is a square matrix itself.  Compute the determinant of each matrix, noting how <acroref type="theorem" acro="SMZD" /> indicates when the matrix is singular or nonsingular.<br /><br />
<acroref type="archetype" acro="A" />,
<acroref type="archetype" acro="B" />,
<acroref type="archetype" acro="F" />,
<acroref type="archetype" acro="K" />,
<acroref type="archetype" acro="L" />
</problem>
</exercise>

<exercise type="M" number="20" rough="matrix of cofactors of 3x3 to get inverse">
<problem contributor="robertbeezer">Construct a $3\times 3$ nonsingular matrix and call it $A$.  Then, for each entry of the matrix, compute the corresponding cofactor, and create a new $3\times 3$ matrix full of these cofactors by placing the cofactor of an entry in the same location as the entry it was based on.  Once complete, call this matrix $C$.  Compute $A\transpose{C}$.  Any observations?  Repeat with a new matrix, or perhaps with a $4\times 4$ matrix.
</problem>
<solution contributor="robertbeezer">The result of these computations should be a matrix with the value of $\detname{A}$ in the diagonal entries and zeros elsewhere.  The suggestion of using a nonsingular matrix was partially so that it was obvious that the value of the determinant appears on the diagonal.<br /><br />
This result (which is true in general) provides a method for computing the inverse of a nonsingular matrix.  Since $A\transpose{C}=\detname{A}I_n$, we can multiply by the reciprocal of the determinant (which is nonzero!) and the inverse of $A$ (it exists!) to arrive at an expression for the matrix inverse:
<equation>
\inverse{A}=\frac{1}{\detname{A}}\transpose{C}
</equation>
</solution>
</exercise>

<exercise type="M" number="30" rough="det(A+B) != det(A) + det(B)">
<problem contributor="robertbeezer">Construct an example to show that the following statement is not true for all square matrices $A$ and $B$ of the same size: $\detname{A+B}=\detname{A}+\detname{B}$.
</problem>
</exercise>

<exercise type="T" number="10" rough="prove nonsing product is built from nonsing pieces">
<problem contributor="robertbeezer"><acroref type="theorem" acro="NPNT" /> says that if the product of square matrices $AB$ is nonsingular, then the individual matrices $A$ and $B$ are nonsingular also.  Construct a new proof of this result making use of theorems about determinants of matrices.
</problem>
</exercise>

<exercise type="T" number="15" rough="Theorem DRCZ as corollary of Theorem DRCM">
<problem contributor="robertbeezer">Use <acroref type="theorem" acro="DRCM" /> to prove <acroref type="theorem" acro="DZRC" /> as a corollary.  (See <acroref type="technique" acro="LC" />.)
</problem>
</exercise>

<exercise type="T" number="20" rough="det(cA) = c^n det(A)">
<problem contributor="robertbeezer">Suppose that $A$ is a square matrix of size $n$ and $\alpha\in\complex{\null}$ is a scalar.  Prove that $\detname{\alpha A}=\alpha^n\detname{A}$.
</problem>
</exercise>

<exercise type="T" number="25" rough="Column half of proof of Theorem DRCM via transpose">
<problem contributor="robertbeezer">Employ <acroref type="theorem" acro="DT" /> to construct the second half of the proof of <acroref type="theorem" acro="DRCM" /> (the portion about a multiple of a column).
</problem>
</exercise>

</exercisesubsection>

</section>