Question

If $x,y,z$ are non zero real numbers, then the inverse of matrix $A=$ $\left[\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right]$ is

• A. $\left[\begin{array}{ccc}{x}^{-1}& 0& 0\\ 0& y^{-1}& 0\\ 0& 0& z^{-1}\end{array}\right]$
• B. $xyz\left[\begin{array}{ccc}{x}^{-1}& 0& 0\\ 0& y^{-1}& 0\\ 0& 0& z^{-1}\end{array}\right]$
• C. $\frac{1}{xyz}\left[\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right]$
• D. $\frac{1}{xyz}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Answer 1

The correct option is A $\left[\begin{array}{ccc}{x}^{-1}& 0& 0\\ 0& y^{-1}& 0\\ 0& 0& z^{-1}\end{array}\right]$

$A=\left[\begin{array}{ccc}x& 0& 0\\ 0& y& 0\\ 0& 0& z\end{array}\right]$

Here, $|A|=x(yz-0)+0+0$

$\Rightarrow |A|=xyz$

Since, $x,y,z$ are non-zero real numbers.

So, $A^{-1}$ exists.

$A^{-1}=\frac{adjA}{|A|}$ and $adjA=C^T$

$C_{11}=(-1{)}^{1+1}\left|\begin{array}{cc}y& 0\\ 0& z\end{array}\right|$

$\Rightarrow C_{11}=yz$

$C_{12}=(-1{)}^{1+2}\left|\begin{array}{cc}0& 0\\ 0& z\end{array}\right|$

$\Rightarrow C_{12}=0$

$C_{13}=(-1{)}^{1+3}\left|\begin{array}{cc}0& 2y\\ 0& 0\end{array}\right|$

$\Rightarrow C_{13}=0$

$C_{21}=(-1{)}^{2+1}\left|\begin{array}{cc}0& 0\\ 0& z\end{array}\right|$

$\Rightarrow C_{21}=0$

$C_{22}=(-1{)}^{2+2}\left|\begin{array}{cc}x& 0\\ 0& z\end{array}\right|$

$\Rightarrow C_{22}=xz-0=xz$

$C_{23}=(-1{)}^{2+3}\left|\begin{array}{cc}x& 0\\ 0& 0\end{array}\right|$

$\Rightarrow C_{23}=0$

$C_{31}=(-1{)}^{3+1}\left|\begin{array}{cc}0& 0\\ y& 0\end{array}\right|$

$\Rightarrow C_{31}=0$

$C_{32}=(-1{)}^{3+2}\left|\begin{array}{cc}x& 0\\ 0& 0\end{array}\right|$

$\Rightarrow C_{32}=0$

$C_{33}=(-1{)}^{3+3}\left|\begin{array}{cc}x& 0\\ 0& y\end{array}\right|$

$\Rightarrow C_{33}=xy$

Hence, the co-factor matrix is $C=\left[\begin{array}{ccc}yz& 0& 0\\ 0& xz& 0\\ 0& 0& xy\end{array}\right]$

$\Rightarrow adjA=C^T=\left[\begin{array}{ccc}yz& 0& 0\\ 0& xz& 0\\ 0& 0& xy\end{array}\right]$

$\Rightarrow A^{-1}=\frac{adjA}{|A|}=\frac{1}{xyz}\left[\begin{array}{ccc}yz& 0& 0\\ 0& xz& 0\\ 0& 0& xy\end{array}\right]$

$\Rightarrow A^{-1}=\left[\begin{array}{ccc}\frac{1}{x}& 0& 0\\ 0& \frac{1}{y}& 0\\ 0& 0& \frac{1}{z}\end{array}\right]$