Chapter 7.7, Problem 19E

a.

To determine

To find the minors of the matrix.

Answer to Problem 19E

The minors of thematrix are obtained as,

  $\begin{array}{l}{\text{M}}_{11}=-5{\text{ M}}_{12}=2\\ {\text{M}}_{21}=4{\text{ M}}_{22}=3\end{array}$

Explanation of Solution

Given information :

The matrixis given as,

  $\text{A}=\left[\begin{array}{ccc}-4& 6& 3\\ 7& -2& 8\\ 1& 0& -5\end{array}\right]$

Calculation :

The minors of the matrix can be obtainedas,

For $M_{11}$ , delete the first row and the first column of A and evaluate.

  $\begin{array}{l}{M}_{11}=\left|\begin{array}{cc}-2& 5\\ 0& -5\end{array}\right|\\ M_{11}=10\end{array}$

Continuing the pattern; other minors can be obtained as,

  $\begin{array}{l}{M}_{12}=-43M_{13}=2\\ M_{21}=-30M_{22}=17M_{23}=-6\\ M_{31}=54M_{32}=-53M_{33}=-34\end{array}$

Therefore,

The minors of thematrix are obtained as,

  $\begin{array}{l}{M}_{11}=10M_{12}=-43M_{13}=2\\ M_{21}=-30M_{22}=17M_{23}=-6\\ M_{31}=54M_{32}=-53M_{33}=-34\end{array}$

b.

To determine

To find the cofactors of the matrix.

Answer to Problem 19E

The cofactors of the matrix are obtained as,

  $\begin{array}{l}{C}_{11}=10C_{12}=43C_{13}=2\\ C_{21}=30C_{22}=17C_{23}=6\\ C_{31}=54C_{32}=53C_{33}=-34\end{array}$

Explanation of Solution

Given information :

The matrixis given as,

  $\text{A}=\left[\begin{array}{cc}3& 4\\ 2& -5\end{array}\right]$

Calculation :

To find cofactors, combine the minors with the sign pattern of the cofactors.

The cofactors of the matrix can be obtained as,

  $\begin{array}{l}{C}_{11}=10C_{12}=43C_{13}=2\\ C_{21}=30C_{22}=17C_{23}=6\\ C_{31}=54C_{32}=53C_{33}=-34\end{array}$

Therefore,

The cofactors of the matrix are obtained as,

  $\begin{array}{l}{C}_{11}=10C_{12}=43C_{13}=2\\ C_{21}=30C_{22}=17C_{23}=6\\ C_{31}=54C_{32}=53C_{33}=-34\end{array}$