Chapter 8.4, Problem 37E

To determine

To calculate: Find all the minors and cofactors of the given matrix.

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

Answer to Problem 37E

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

Explanation of Solution

Concept used:

Consider a $3\times 3$ matrix A, such that $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$

Minor of $a_{11}=M_{11}=\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|$

Minor of $a_{12}=M_{12}=\left|\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right|$

Minor of $a_{13}=M_{13}=\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|$

Minor of $a_{21}=M_{21}=\left|\begin{array}{cc}{a}_{12}& a_{13}\\ a_{32}& a_{33}\end{array}\right|$

Minor of $a_{22}=M_{22}=\left|\begin{array}{cc}{a}_{11}& a_{13}\\ a_{31}& a_{33}\end{array}\right|$

Minor of $a_{23}=M_{23}=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{31}& a_{32}\end{array}\right|$

Minor of $a_{31}=M_{31}=\left|\begin{array}{cc}{a}_{12}& a_{13}\\ a_{22}& a_{23}\end{array}\right|$

Minor of $a_{32}=M_{32}=\left|\begin{array}{cc}{a}_{11}& a_{13}\\ a_{21}& a_{23}\end{array}\right|$

Minor of $a_{33}=M_{33}=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|$

Cofactor of $a_{ij}=C_{ij}={\left(-1\right)}^{i+j}{M}_{ij}$

Calculation:

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

Minor $M_{11}=\left|\begin{array}{cc}-2& 8\\ 0& -5\end{array}\right|=\left(-2\right)\left(-5\right)-0\left(8\right)=10$

Cofactor $C_{11}={\left(-1\right)}^{1+1}{M}_{11}=10$

Minor $M_{12}=\left|\begin{array}{cc}7& 8\\ 1& -5\end{array}\right|=\left(7\right)\left(-5\right)-8\left(1\right)=-43$

Cofactor $C_{12}={\left(-1\right)}^{1+2}{M}_{12}=43$

Minor $M_{13}=\left|\begin{array}{cc}7& -2\\ 1& 0\end{array}\right|=\left(7\right)\left(0\right)-2\left(1\right)=-2$

Cofactor $C_{13}={\left(-1\right)}^{1+3}{M}_{13}=2$

Minor $M_{21}=\left|\begin{array}{cc}6& 3\\ 0& -5\end{array}\right|=\left(6\right)\left(-5\right)-0\left(3\right)=-30$

Cofactor $C_{21}={\left(-1\right)}^{2+1}{M}_{21}=30$

Minor $M_{22}=\left|\begin{array}{cc}-4& 3\\ 1& -5\end{array}\right|=\left(-4\right)\left(-5\right)-3\left(1\right)=17$

Cofactor $C_{22}={\left(-1\right)}^{2+2}{M}_{22}=17$

Minor $M_{23}=\left|\begin{array}{cc}-4& 6\\ 1& 0\end{array}\right|=\left(-4\right)\left(0\right)-1\left(6\right)=-6$

Cofactor $C_{23}={\left(-1\right)}^{2+3}{M}_{23}=6$

Minor $M_{31}=\left|\begin{array}{cc}6& 3\\ -2& 8\end{array}\right|=\left(6\right)\left(8\right)-3\left(-2\right)=54$

Cofactor $C_{31}={\left(-1\right)}^{3+1}{M}_{31}=54$

Minor $M_{32}=\left|\begin{array}{cc}-4& 3\\ 7& 8\end{array}\right|=\left(-4\right)\left(8\right)-3\left(7\right)=-53$

Cofactor $C_{32}={\left(-1\right)}^{3+2}{M}_{32}=53$

Minor $M_{33}=\left|\begin{array}{cc}-4& 6\\ 7& -2\end{array}\right|=\left(-4\right)\left(-2\right)-6\left(7\right)=-34$

Cofactor $C_{33}={\left(-1\right)}^{3+3}{M}_{33}=-34$