Question

Given $x-y+3z=5$; $4x+2y-z=0$; $-x+3y+z=5$;

If

A=$\left[\begin{array}{ccc}1& -1& 3\\ 4& 2& -1\\ -1& 3& 1\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],D=\left[\begin{array}{c}5\\ 0\\ 5\end{array}\right]$ such that AX=D.

Show that $A$ is non singular and the cofactor elements of a matrix $A$ is $\left[\begin{array}{ccc}+(2+3)& -(4-1)& +(12+2)\\ -(-1-9)& +(1+3)& -(3-1)\\ +(1-6)& -(-1-12)& +(2+4)\end{array}\right]$

Answer 1

Given $A=\left[\begin{array}{ccc}1& -1& 3\\ 4& 2& -1\\ -1& 3& 1\end{array}\right]$

$\mid A\mid =\left|\begin{array}{ccc}1& -1& 3\\ 4& 2& -1\\ -1& 3& 1\end{array}\right|=1\left(5\right)-1\left(3\right)+3\left(14\right)=44$

$A$ is non singular

let element of cofactor matrix is $a_{ij}$

$a_{11}=(-1{)}^{1+1}\left|\begin{array}{cc}2& -1\\ 3& 1\end{array}\right|=+(2+3)$

$a_{12}=(-1{)}^{1+2}\left|\begin{array}{cc}4& -1\\ -1& 1\end{array}\right|=-(4-1)$

$a_{13}=(-1{)}^{1+3}\left|\begin{array}{cc}4& 2\\ -1& 3\end{array}\right|=+(12+2)$

$a_{21}=(-1{)}^{2+1}\left|\begin{array}{cc}-1& 3\\ 3& 1\end{array}\right|=-(-1-9)$

$a_{22}=(-1{)}^{2+2}\left|\begin{array}{cc}1& 3\\ -1& 1\end{array}\right|=+(1+3)$

$a_{23}=(-1{)}^{2+3}\left|\begin{array}{cc}1& -1\\ -1& 3\end{array}\right|=-(3-1)$

$a_{31}=(-1{)}^{3+1}\left|\begin{array}{cc}-1& 3\\ 2& -1\end{array}\right|=+(1-6)$

$a_{32}=(-1{)}^{3+2}\left|\begin{array}{cc}1& 3\\ 4& -1\end{array}\right|=-(-1-12)$

$a_{33}=(-1{)}^{+3}\left|\begin{array}{cc}1& -1\\ 4& 2\end{array}\right|=+(2+4)$

$\therefore$ cofactor matrix is $\left[\begin{array}{ccc}+(2+3)& -(4-1)& +(12+2)\\ -(-1-9)& +(1+3)& -(3-1)\\ +(1-6)& -(-1-12)& +(2+4)\end{array}\right]$