Question

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

• A. $ \begin{bmatrix} \frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & -\frac{1}{7} \\ \end{bmatrix} $
• B. $ \begin{bmatrix} -\frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & -7 \\ \end{bmatrix} $
• C. $ \begin{bmatrix} \frac{1}{3} & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & 7 \\ \end{bmatrix}$
• D. $ \begin{bmatrix} \frac{1}{3} & 0 & 0 \\ 0 & -\frac{1}{2} & 0 \\ 0 & 0 & 7 \\ \end{bmatrix} $

Answer 1

Answer: (A)

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