Question

The number of $2 \times 2$ matrices $A=\begin{bmatrix}a & b \\ c & d\end{bmatrix}$ for which $\begin{bmatrix}a & b \\c & d\end{bmatrix}^{-1}=\begin{bmatrix}1 / a & 1 / b \\1 / c & 1 / d\end{bmatrix},(a, b, c, d \in R ) \text { is }$

• A. 0
• B. 1
• C. 2
• D. infinite

Answer 1

Answer: (A)

If $a d-b c \neq 0$, then
$A^{-1}=\frac{1}{a d-b c}\begin{vmatrix}d & -b \\-c & a\end{vmatrix}$
Thus, $\frac{d}{a d-b c} =\frac{1}{a} \Leftrightarrow a d=a d-b c $
$\Leftrightarrow b c=0 \Leftrightarrow b=0 \text { or } c=0$
Therefore,
$\begin{vmatrix}a & b \\c & d\end{vmatrix}^{-1}=\begin{vmatrix}1 / a & 1 / b \\1 / c & 1 / d\end{vmatrix}$
is never possible.