Let A be a 3 × 3 real matrix such that A 1 0 1 = 2 1 0 1 , A − 1 0 1 = 4 − 1 0 1 , A 0 1 0 = 2 0 1 0 . Then,…

Let, $A=\left[\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ x_3& y_3& z_3\end{array}\right]$

It is given that, $A\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]=\left[\begin{array}{c}2\\ 0\\ 2\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}{x}_1+z_1\\ x_2+z_2\\ x_3+z_3\end{array}\right]=\left[\begin{array}{c}2\\ 0\\ 2\end{array}\right]$

$\Rightarrow x_1+z_1=2, x_2+z_2=0, x_3+z_3=0 ...\left(i\right)$

Also, $A\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right]=\left[\begin{array}{c}-4\\ 0\\ 4\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}-x_1+z_1\\ -x_2+z_2\\ -x_3+z_3\end{array}\right]=\left[\begin{array}{c}-4\\ 0\\ 4\end{array}\right]$

$\Rightarrow -x_1+z_1=-4, -x_2+z_2=0, -x_3+z_3=4 ...\left(ii\right)$

Now, $A\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 2\\ 0\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]=\left[\begin{array}{c}0\\ 2\\ 0\end{array}\right]$

$\Rightarrow y_1=0, y_2=2, y_3=0 ...\left(iii\right)$

Using $\left(i\right) \text{and} \left(ii\right)$,

$\Rightarrow x_1=3, x_2=0, x_3=-1$

$\Rightarrow z_1=-1, z_2=0, z_3=3$

$\therefore A=\left[\begin{array}{ccc}3& 0& -1\\ 0& 2& 0\\ -1& 0& 3\end{array}\right]$

Now, $\left(A-3I\right)\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-1\\ 2\\ 3\end{array}\right]$

$\Rightarrow \left[\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-1\\ 2\\ 3\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}-z\\ -y\\ -x\end{array}\right]=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$

$\Rightarrow z=-1, y=-2, x=-3$