Question

If $A$ is a $ 2\times 2 $ matrix and $ |A|=2, $ then the matrix represented by $A (adj A)$ is equal to

• A. $ \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] $
• B. $ \left[ \begin{matrix} 2 & 0 \\ 0 & 2 \\ \end{matrix} \right] $
• C. $ \left[ \begin{matrix} 1/2 & 0 \\ 0 & 1/2 \\ \end{matrix} \right] $
• D. $ \left[ \begin{matrix} 0 & 2 \\ 2 & 0 \\ \end{matrix} \right] $

Answer 1

Answer: (B)

Consider a $ 2\times 2 $ matrix whose
$ |A|=2 $ $ A=\left[ \frac{4}{2}\,\,\frac{1}{1} \right] $
Now, $ adj\,\,A=\left[ \begin{matrix} 1 & -2 \\ -1 & 4 \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & -1 \\ -2 & 4 \\ \end{matrix} \right] $
$ \Rightarrow $ $ A\,\,(adj\,\,A)=\left[ \begin{matrix} 4 & 1 \\ 2 & 1 \\ \end{matrix} \right]\,\,\left[ \begin{matrix} 1 & -1 \\ -2 & 4 \\ \end{matrix} \right] $
$ =\left[ \begin{matrix} 4-2 & -4+4 \\ 2-2 & -2+4 \\ \end{matrix} \right] $
$ =\left[ \begin{matrix} 2 & 0 \\ 0 & 2 \\ \end{matrix} \right] $