If $A X=D$ represents the system of simultaneous linear equations $x+y+z=6$, $5 x-y+2 z=3$ and $2 x+y-z=-5$, then $(\operatorname{Adj} A) D=$

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Aicharya · Accepted Answer

The given system of equation is $5 x-y+2 z=3, x+y+z=6$ and $2 x+y-z=-5$ In matrix form, if may be represented as where $A=\left[\begin{array}{ccc}1 & 1 & 1 \ 5 & -1 & 2 \ 2 & 1 & -1\end{array}ight], x=\left[\begin{array}{l}x \ y \ z\end{array}ight]$ and $D=\left[\begin{array}{c}6 \ 3 \ -5\end{array}ight]$ Thus, $A$ is a non-singular matrix, so $A^{-1}$ exist. $\begin{aligned} & C_{11}=(-1)^{1+1}\left|\begin{array}{cc}-1 & 2 \ 1 & -1\end{array}ight|=-1 \ & C_{12}=(-1)^{1+2}\left|\begin{array}{cc}5 & 2 \ 2 & -1\end{array}ight|=9 \ & C_{13}=(-1)^{1+3}\left|\begin{array}{cc}5 & -1 \ 2 & 1\end{array}ight|=7 \ & C_{21}=(-1)^{2+1}\left|\begin{array}{cc}1 & 1 \ 1 & -1\end{array}ight|=2\end{aligned}$ $\begin{aligned} & C_{22}=(-1)^{2+2}\left|\begin{array}{ll}1 & 1 \ 2 & -1\end{array}ight|=-3 \ & C_{23}=(-1)^{2+3}\left|\begin{array}{ll}1 & 1 \ 2 & 1\end{array}ight|=1 \ & C_{31}=(-1)^{3+1}\left|\begin{array}{cc}1 & 1 \ -1 & 2\end{array}ight|=3 \ & C_{32}=(-1)^{3+2}\left|\begin{array}{ll}1 & 1 \ 5 & 2\end{array}ight|=3 \ & C_{33}=(-1)^{3+3}\left|\begin{array}{cc}1 & 1 \ 5 & -1\end{array}ight|=-6\end{aligned}$ Then, $\operatorname{adj}(A)=\left[\begin{array}{ccc}-1 & 9 & 7 \ 2 & -3 & 1 \ 3 & 3 & -6\end{array}ight]^T=\left[\begin{array}{ccc}-1 & 2 & 3 \ 9 & -3 & 3 \ 7 & 1 & -6\end{array}ight]$ $(\operatorname{adj} A) \cdot D=\left[\begin{array}{ccc}-1 & 2 & 3 \ 9 & -3 & 3 \ 7 & 1 & -6\end{array}ight]\left[\begin{array}{c}6 \ 3 \ -5\end{array}ight]$ $=\left[\begin{array}{c}-6+6-15 \ 54-9-15 \ 42+3+30\end{array}ight]=\left[\begin{array}{c}-15 \ 30 \ 75\end{array}ight]$

If $A X=D$ represents the system of simultaneous linear equations $x+y+z=6$, $5 x-y+2 z=3$ and $2 x+y-z=-5$,…

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