Question

If $X$ and $Y$ are two non-singular matrices such that $XY{X}^{-1}=\left(\begin{array}{ccc}3& 0& 0\\ 0& 2& 0\\ 0& 0& -7\end{array}\right),$ then $X{Y}^{-1}{X}^{-1}$ is equal to

• A. $\left[\begin{array}{ccc}\frac{1}{3}& 0& 0\\ 0& \frac{1}{2}& 0\\ 0& 0& -\frac{1}{7}\end{array}\right]$
• B. $\left[\begin{array}{ccc}-\frac{1}{3}& 0& 0\\ 0& \frac{1}{2}& 0\\ 0& 0& -7\end{array}\right]$
• C. $\left[\begin{array}{ccc}\frac{1}{3}& 0& 0\\ 0& \frac{1}{2}& 0\\ 0& 0& 7\end{array}\right]$
• D. $\left[\begin{array}{ccc}\frac{1}{3}& 0& 0\\ 0& -\frac{1}{2}& 0\\ 0& 0& 7\end{array}\right]$

Answer 1

Answer: (A)

Given, $XY{X}^{-1}=\left[\begin{array}{ccc}3& 0& 0\\ 0& 2& 0\\ 0& 0& -7\end{array}\right]$
Taking inverse both the sides, we get
${(XY{X}^{-1})}^{-1}={\left[\begin{array}{ccc}3& 0& 0\\ 0& 2& 0\\ 0& 0& -7\end{array}\right]}^{-1}$
$\Rightarrow$ $X{Y}^{-1}{X}^{-1}={\left[\begin{array}{ccc}3& 0& 0\\ 0& 2& 0\\ 0& 0& -7\end{array}\right]}^{-1}$
Now, we are to determine inverse of $\left[\begin{array}{ccc}3& 0& 0\\ 0& 2& 0\\ 0& 0& -7\end{array}\right]=A(consider)$
adj $A=\left[\begin{array}{ccc}-14& 0& 0\\ 0& -21& 0\\ 0& 0& 6\end{array}\right]$
and $|A|=-42$ $A^{-1}=\frac{1}{|A|}(adjA)=\left[\begin{array}{ccc}1/3& 0& 0\\ 0& 1/2& 0\\ 0& 0& -1/7\end{array}\right]$
Hence, $X{Y}^{-1}{X}^{-1}=\left[\begin{array}{ccc}1/3& 0& 0\\ 0& 1/2& 0\\ 0& 0& -1/7\end{array}\right]$