Chapter 8, Problem 115RE

a.

To determine

The minors of the matrix $\left[\begin{array}{ccc}3& 2& -1\\ -2& 5& 0\\ 1& 8& 6\end{array}\right]$ .

Answer to Problem 115RE

The Minors of the given matrix are $M_{11}=30,M_{12}=-12,M_{13}=-21,M_{21}=20,M_{22}=19,$

  $M_{23}=22,M_{31}=5,M_{32}=-2,M_{33}=19.$

Explanation of Solution

Given information :

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

The minor of an element $a_{11}$ of a matrix $\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$ is $M_{11}=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|\left(\text{delete}\text{first}\text{row}\text{and}\text{first}\text{column}\right)$

Similar way, minor of $3$ is $M_{11}=\left|\begin{array}{cc}5& 0\\ 8& 6\end{array}\right|=30$

  minor of $2$ is $M_{12}=\left|\begin{array}{cc}-2& 0\\ 1& 6\end{array}\right|=-12$

  minor of $-1$ is $M_{13}=\left|\begin{array}{cc}-2& 5\\ 1& 8\end{array}\right|=-16-5=-21$

  minor of $-2$ is $M_{21}=\left|\begin{array}{cc}2& -1\\ 8& 6\end{array}\right|=12+8=20$

  minor of $5$ is $M_{22}=\left|\begin{array}{cc}3& -1\\ 1& 6\end{array}\right|=18+1=19$

  minor of $0$ is $M_{23}=\left|\begin{array}{cc}3& 2\\ 1& 8\end{array}\right|=24-2=22$

  minor of $1$ is $M_{31}=\left|\begin{array}{cc}2& -1\\ 5& 0\end{array}\right|=0+5=5$

  minor of $8$ is $M_{32}=\left|\begin{array}{cc}3& -1\\ -2& 0\end{array}\right|=0-2=-2$

  minor of $6$ is $M_{33}=\left|\begin{array}{cc}3& 2\\ -2& 5\end{array}\right|=15+4=19$

Hence, the minors of the given matrix are $M_{11}=30,M_{12}=-12,M_{13}=-21,M_{21}=20,M_{22}=19,$

  $M_{23}=22,M_{31}=5,M_{32}=-2,M_{33}=19.$

b.

To determine

The cofactors of the matrix $\left[\begin{array}{ccc}3& 2& -1\\ -2& 5& 0\\ 1& 8& 6\end{array}\right]$ .

Answer to Problem 115RE

The Co-factors of the given matrix are $C_{11}=30,C_{12}=12,C_{13}=-21,M_{21}=-20,C_{22}=19,$

  $C_{23}=-22,C_{31}=5,C_{32}=2,C_{33}=19.$

Explanation of Solution

Given information :

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

The minor of an element $a_{11}$ of a matrix $\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$ is $C_{11}={\left(-1\right)}^{1+1}{M}_{11}=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|\left(\text{delete}\text{first}\text{row}\text{and}\text{first}\text{column}\right)$

Similar way, cofactor of $3$ is $C_{11}={\left(-1\right)}^{1+1}{M}_{11}=\left|\begin{array}{cc}5& 0\\ 8& 6\end{array}\right|=30$

  cofactor of $2$ is $C_{12}={\left(-1\right)}^{1+2}{M}_{12}=-\left|\begin{array}{cc}-2& 0\\ 1& 6\end{array}\right|=-12=12$

  cofactor of $-1$ is $C_{13}={\left(-1\right)}^{1+3}{M}_{13}=\left|\begin{array}{cc}-2& 5\\ 1& 8\end{array}\right|=\left(-16-5\right)=-21$

  cofactor of $-2$ is $C_{21}={\left(-1\right)}^{2+1}{M}_{21}=-\left|\begin{array}{cc}2& -1\\ 8& 6\end{array}\right|=-\left(12+8\right)=-20$

  cofactor of $5$ is $C_{22}={\left(-1\right)}^{2+2}{M}_{22}=\left|\begin{array}{cc}3& -1\\ 1& 6\end{array}\right|=18+1=19$

  cofactor of $0$ is $C_{23}={\left(-1\right)}^{2+3}{M}_{23}=-\left|\begin{array}{cc}3& 2\\ 1& 8\end{array}\right|=-\left(24-2\right)=-22$

  cofactor of $1$ is $C_{31}={\left(-1\right)}^{3+1}{M}_{31}=\left|\begin{array}{cc}2& -1\\ 5& 0\end{array}\right|=0+5=5$

  cofactor of $8$ is $C_{32}={\left(-1\right)}^{3+2}{M}_{32}=-\left|\begin{array}{cc}3& -1\\ -2& 0\end{array}\right|=-\left(0-2\right)=2$

  cofactor of $6$ is $C_{33}={\left(-1\right)}^{3+3}{M}_{33}=\left|\begin{array}{cc}3& 2\\ -2& 5\end{array}\right|=15+4=19$

Hence, the cofactors of the given matrix are $C_{11}=30,C_{12}=12,C_{13}=-21,M_{21}=-20,C_{22}=19,$

  $C_{23}=-22,C_{31}=5,C_{32}=2,C_{33}=19.$