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Section 4.2 Minors and cofactors

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 \(n=3.\)

We focus on a particular element \(a_{i,j}\) of a square matrix \(A\text{.}\)

Definition 4.2.1. The \(i,j\) minor of a matrix \(A\).

\(M_{i,j}\text{,}\) the \(i,j\) minor of a matrix \(A\) is computed by deleting the row and column containing \(a_{i,j}\) and evaluating the determinant of what remains.

\begin{equation*} \qquad\qquad\qquad\downarrow \text{ column } C_j \text{ deleted}\\ 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{equation*}
Definition 4.2.2. The \(i,j\) cofactor of a matrix \(A\).

\(C_{i,j}\text{,}\) the \(i,j\) cofactor of a matrix \(A\text{,}\) satisfies

\begin{equation*} C_{i,j}=(-1)^{i+j}M_{i,j} \end{equation*}

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.

Example 4.2.3. A cofactor of a matrix.

We find \(M_{2,3}\) and \(C_{2,3}\) for the matrix

\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*}

When we delete row \(R_2\) and column \(C_3\) from \(A\) we get the matrix

\begin{equation*} \begin{bmatrix} 1\amp2\amp3\\ 3\amp4\amp-1\\ 1\amp2\amp1 \end{bmatrix} \end{equation*}

We have already calculated the determinant of this matrix in Example 4.1.2 to be \(4,\) so \(M_{2,3}=4\) and \(C_{2,3}=(-1)^5 4=-4.\)

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

\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{4.2.1} \end{equation}

In other words, the pattern of \(1\) and \(-1\) is like a checkerboard pattern of light squares and dark squares.

Since the minor \(M_{i,j}\) is simply a number, we may form an new matrix called the matrix of minors \(M\) whose entries are minors. Similarly since the cofactor \(C_{i,j}\) is simply a number, we may form an new matrix called the cofactor matrix whose entries are cofactors.

Definition 4.2.4. Cofactor matrix and the matrix of minors.

The matrix of minors \(M\) is defined by

\begin{equation*} M=[M_{i,j}] \end{equation*}

and the cofactor matrix \(C\) is defined by

\begin{equation*} C=[C_{i,j}] \end{equation*}
Example 4.2.5. Matrix of minors.

We will compute the matrix of minors for the matrix

\begin{equation*} \begin{bmatrix} 1\amp 2\amp3\\ 3\amp 4\amp -1\\ 1\amp 2\amp 1 \end{bmatrix} \end{equation*}

From the definition of the matrix of minors:

\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*}
Example 4.2.6. Cofactor matrix.

We will compute the cofactor matrix of the matrix

\begin{equation*} \begin{bmatrix} 1\amp 2\amp3\\ 3\amp 4\amp -1\\ 1\amp 2\amp 1 \end{bmatrix} \end{equation*}

From the definition of the cofactor matrix:

\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*}
Observation 4.2.7. Minors, cofactors and the checkerboard pattern.

The answers from Example 4.2.5 and Example 4.2.6 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 \(P\) (4.2.1) 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.

Suppose a matrix \(A\) has matrix of minors

\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*}

What is \(C\text{,}\) the cofactor matrix of \(A\text{?}\)

Answer
\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*}