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<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="section-20"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">3.2</span> <span class="title">Minors and cofactors</span>
</h2>
<p id="p-534">The evaluation of the determinant by cofactors changes the problem of finding the determinant of a matrix of order \(n\) to that of finding the determinant several matrices of order \(n-1.\) So far, we know how to evaluate the determinant of a matrix of order \(n\) when \(n\leq 3.\) This method will, then, allow us to evaluate the determinant of a matrix of order \(n=4\) by reducing the problem to solving the determinant of several matrices of order \(n=3.\)</p>
<p id="p-535">We focus on a particular element \(a_{i,j}\) of a square matrix \(A\text{.}\)</p>
<article class="definition definition-like" id="MatrixMinorDefinition"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">3.2.1</span><span class="period">.</span><span class="space"> </span><span class="title">The \(i,j\) minor of a matrix \(A\).</span>
</h6>
<p id="p-536">\(M_{i,j}\text{,}\) the <dfn class="terminology">\(i,j\) minor</dfn> of a matrix \(A\) is computed by deleting the row and column containing \(a_{i,j}\) and evaluating the determinant of what remains.</p>
<div class="displaymath">
\begin{align*}
\downarrow\text{ column }C_j\text{ deleted}\ \quad\qquad\qquad\\
M_{i,j}=\det
\left[\begin{array}{c|c|c}
{}******\amp \amp *****\\
{}******\amp \amp *****\\
{}******\amp \amp *****\\\hline
\cdots\amp a_{i,j}\amp\cdots\\ \hline
{}******\amp \amp *****\\
{}******\amp \amp *****\\
{}******\amp \amp *****
\end{array}\right]
\gets \text{ row } R_i \text{ deleted}
\end{align*}
</div></article><article class="definition definition-like" id="MatrixCofactorDefinition"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">3.2.2</span><span class="period">.</span><span class="space"> </span><span class="title">The \(i,j\) cofactor of a matrix \(A\).</span>
</h6>
<p id="p-537">\(C_{i,j}\text{,}\) the <dfn class="terminology">\(i,j\) cofactor of a matrix</dfn> \(A\text{,}\) satisfies</p>
<div class="displaymath">
\begin{equation*}
C_{i,j}=(-1)^{i+j}M_{i,j}
\end{equation*}
</div></article><p id="p-538">In other words, \(C_{i,j}=\pm M_{i,j}\text{,}\) with the sign being \(+\) if \(i+j\) is even and \(-\) if \(i+j\) is odd.</p>
<article class="example example-like" id="example-25"><a data-knowl="" class="id-ref example-knowl original" data-refid="hk-example-25"><h6 class="heading">
<span class="type">Example</span><span class="space"> </span><span class="codenumber">3.2.3</span><span class="period">.</span><span class="space"> </span><span class="title">A cofactor of a matrix.</span>
</h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-example-25"><article class="example example-like"><p id="p-539">We find \(M_{2,3}\) and \(C_{2,3}\) for the matrix</p>
<div class="displaymath">
\begin{equation*}
A=
\begin{bmatrix}1\amp2\amp2\amp3\\
-1\amp4\amp5\amp3\\
3\amp4\amp8\amp-1\\
1\amp2\amp2\amp1 \end{bmatrix}
\end{equation*}
</div>
<p class="continuation">When we delete row \(R_2\) and column \(C_3\) from \(A\) we get the matrix</p>
<div class="displaymath">
\begin{equation*}
\begin{bmatrix}
1\amp2\amp3\\
3\amp4\amp-1\\
1\amp2\amp1
\end{bmatrix}
\end{equation*}
</div>
<p class="continuation">We have already calculated the determinant of this matrix in <a class="xref" data-knowl="./knowl/det_example1.html" title="Example 3.1.2: Determinants of small matrices">Example 3.1.2</a> to be \(4,\) so \(M_{2,3}=4\) and \(C_{2,3}=(-1)^5 4=-4.\)</p></article></div>
<p id="p-540">There is a nice way of visualizing the pattern of \((-1)^{i+j}.\) Consider the matrix \(P=[p_{i,j}]\) where \(p_{i,j}=(-1)^{i+j}\text{.}\) The next entry to the right of \(p_{i,j}\) is \(p_{i,j+1}\text{,}\) so that the exponent of \(-1\) is increased by one. Hence if \(p_{i,j}=1\) then \(p_{i,j+1}=-1\) and \(p_{i,j}=-1\) then \(p_{i,j+1}=1\text{.}\) This means that the entries in a row alternate between \(1\) and \(-1\text{.}\) By an analogous argument, the columns also alternate between \(1\) and \(-1\text{.}\) The upper left entry is \(-1^{1+1}=1\text{,}\) and so the whole matrix is determined. It looks like</p>
<div class="displaymath">
\begin{equation}
P=
\begin{bmatrix}
+1\amp-1\amp+1\amp-1\amp+1\cdots \\
-1\amp+1\amp-1\amp+1\amp-1\cdots \\
+1\amp-1\amp+1\amp-1\amp+1\cdots \\
-1\amp+1\amp-1\amp+1\amp-1\cdots \\
+1\amp-1\amp+1\amp-1\amp+1\cdots \\
\amp\amp\vdots
\end{bmatrix}\label{CheckerboardMatrix}\tag{3.2.1}
\end{equation}
</div>
<p class="continuation">In other words, the pattern of \(1\) and \(-1\) is like a checkerboard pattern of light squares and dark squares.</p>
<p id="p-541">Since the minor \(M_{i,j}\) is simply a number, we may form an new matrix called the <dfn class="terminology">matrix of minors</dfn> \(M\) whose entries are minors. Similarly since the cofactor \(C_{i,j}\) is simply a number, we may form an new matrix called the <em class="emphasis">cofactor matrix</em> whose entries are cofactors.</p>
<article class="definition definition-like" id="definition-33"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">3.2.4</span><span class="period">.</span><span class="space"> </span><span class="title">Cofactor matrix and the matrix of minors.</span>
</h6>
<p id="p-542">The <em class="emphasis">matrix of minors</em> \(M\) is defined by</p>
<div class="displaymath">
\begin{equation*}
M=[M_{i,j}]
\end{equation*}
</div>
<p class="continuation">and the <em class="emphasis">cofactor matrix</em> \(C\) is defined by</p>
<div class="displaymath">
\begin{equation*}
C=[C_{i,j}]
\end{equation*}
</div></article><article class="example example-like" id="MatrixMinorsExmaple"><a data-knowl="" class="id-ref example-knowl original" data-refid="hk-MatrixMinorsExmaple"><h6 class="heading">
<span class="type">Example</span><span class="space"> </span><span class="codenumber">3.2.5</span><span class="period">.</span><span class="space"> </span><span class="title">Matrix of minors.</span>
</h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-MatrixMinorsExmaple"><article class="example example-like"><p id="p-543">We will compute the matrix of minors for the matrix</p>
<div class="displaymath">
\begin{equation*}
\begin{bmatrix}
1\amp 2\amp3\\
3\amp 4\amp -1\\
1\amp 2\amp 1
\end{bmatrix}
\end{equation*}
</div>
<p class="continuation">From the definition of the matrix of minors:</p>
<div class="displaymath">
\begin{align*}
M \amp=
\begin{bmatrix}
M_{1,1} \amp M_{1,2} \amp M_{1,3} \\
M_{2,1} \amp M_{2,2} \amp M_{2,3} \\
M_{3,1} \amp M_{3,2} \amp M_{3,3}
\end{bmatrix} \\
\amp=
\begin{bmatrix}
\det \begin{bmatrix} 4\amp -1 \\ 2\amp 1 \end{bmatrix}
\amp
\det \begin{bmatrix} 3\amp -1 \\ 1\amp 1 \end{bmatrix}
\amp
\det \begin{bmatrix} 3\amp 4 \\ 1\amp 2 \end{bmatrix}
\\
\det \begin{bmatrix} 2\amp 3 \\ 2\amp 1 \end{bmatrix}
\amp
\det \begin{bmatrix} 1\amp 3 \\ 1\amp 1 \end{bmatrix}
\amp
\det \begin{bmatrix} 1\amp 2 \\ 1\amp 2 \end{bmatrix}
\\
\det \begin{bmatrix} 2\amp 3 \\ 4\amp -1 \end{bmatrix}
\amp
\det \begin{bmatrix} 1\amp 3 \\ 3\amp -1 \end{bmatrix}
\amp
\det \begin{bmatrix} 1\amp 2 \\ 3\amp 4 \end{bmatrix}
\end{bmatrix}\\
\amp=
\begin{bmatrix}
6\amp 4 \amp 2\\
-4\amp -2\amp 0\\
-14 \amp -10 \amp -2
\end{bmatrix}
\end{align*}
</div></article></div>
<article class="example example-like" id="CofactorMatrixExmaple"><a data-knowl="" class="id-ref example-knowl original" data-refid="hk-CofactorMatrixExmaple"><h6 class="heading">
<span class="type">Example</span><span class="space"> </span><span class="codenumber">3.2.6</span><span class="period">.</span><span class="space"> </span><span class="title">Cofactor matrix.</span>
</h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-CofactorMatrixExmaple"><article class="example example-like"><p id="p-544">We will compute the cofactor matrix of the matrix</p>
<div class="displaymath">
\begin{equation*}
\begin{bmatrix}
1\amp 2\amp3\\
3\amp 4\amp -1\\
1\amp 2\amp 1
\end{bmatrix}
\end{equation*}
</div>
<p class="continuation">From the definition of the cofactor matrix:</p>
<div class="displaymath">
\begin{align*}
C \amp=
\begin{bmatrix}
C_{1,1} \amp C_{1,2} \amp C_{1,3} \\
C_{2,1} \amp C_{2,2} \amp C_{2,3} \\
C_{3,1} \amp C_{3,2} \amp C_{3,3}
\end{bmatrix} \\
\amp=
\begin{bmatrix}
\det \begin{bmatrix} 4\amp -1 \\ 2\amp 1 \end{bmatrix}
\amp
-\det \begin{bmatrix} 3\amp -1 \\ 1\amp 1 \end{bmatrix}
\amp
\det \begin{bmatrix} 3\amp 4 \\ 1\amp 2 \end{bmatrix}
\\
-\det \begin{bmatrix} 2\amp 3 \\ 2\amp 1 \end{bmatrix}
\amp
\det \begin{bmatrix} 1\amp 3 \\ 1\amp 1 \end{bmatrix}
\amp
-\det \begin{bmatrix} 1\amp 2 \\ 1\amp 2 \end{bmatrix}
\\
\det \begin{bmatrix} 2\amp 3 \\ 4\amp -1 \end{bmatrix}
\amp
-\det \begin{bmatrix} 1\amp 3 \\ 3\amp -1 \end{bmatrix}
\amp
\det \begin{bmatrix} 1\amp 2 \\ 3\amp 4 \end{bmatrix}
\end{bmatrix}\\
\amp=
\begin{bmatrix}
6\amp -4 \amp 2\\
4\amp -2\amp 0\\
-14 \amp 10 \amp -2
\end{bmatrix}
\end{align*}
</div></article></div>
<article class="observation remark-like" id="MinorsCofactorsCheckerboard"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">3.2.7</span><span class="period">.</span><span class="space"> </span><span class="title">Minors, cofactors and the checkerboard pattern.</span>
</h6>
<p id="p-545">The answers from <a class="xref" data-knowl="./knowl/MatrixMinorsExmaple.html" title="Example 3.2.5: Matrix of minors">Example 3.2.5</a> and <a class="xref" data-knowl="./knowl/CofactorMatrixExmaple.html" title="Example 3.2.6: Cofactor matrix">Example 3.2.6</a> are quite similar, each matrix entry being either identical or multiplied by \(-1\text{.}\) The entries multiplied by \(-1\) are in the same positions the \(-1\) entries of the matrix \(P\) <a class="xref" data-knowl="./knowl/CheckerboardMatrix.html" title="Equation 3.2.1">(3.2.1)</a> with the checkerboard pattern. This is a general pattern: Given the matrix of minors \(M\) the matrix of cofactors \(C\) is then computed by multiplying the entries of \(M\) by \(\pm1\) according to the checkerboard pattern.</p></article><article class="exercise exercise-like" id="exercise-35"><a data-knowl="" class="id-ref exercise-knowl original" data-refid="hk-exercise-35"><h6 class="heading">
<span class="type">Checkpoint</span><span class="space"> </span><span class="codenumber">3.2.8</span><span class="period">.</span>
</h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-exercise-35"><article class="exercise exercise-like"><p id="p-546">Suppose a matrix \(A\) has matrix of minors</p>
<div class="displaymath">
\begin{equation*}
M=
\begin{bmatrix}
1 \amp 3 \amp 4\\
2 \amp -1\amp 2\\
0 \amp -2\amp 5
\end{bmatrix}\text{.}
\end{equation*}
</div>
<p class="continuation">What is \(C\text{,}\) the cofactor matrix of \(A\text{?}\)</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-1" id="answer-1"><span class="type">Answer.</span> </a><div class="hidden-content tex2jax_ignore" id="hk-answer-1"><div class="answer solution-like"><div class="displaymath" id="p-547">
\begin{equation*}
C=
\begin{bmatrix}
1 \amp-3 \amp 4\\
-2 \amp -1\amp-2\\
0 \amp 2\amp 5
\end{bmatrix}\text{.}
\end{equation*}
</div></div></div>
</div></article></div></section></div></main>
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