Question

Let $A$ and $B$ are two non-singular matrices of order $3$ such that $A+B=2I$ and $A^{-1}+B^{-1}=3I$ , then $AB$ is equal to (where, $I$ is the identity matrix of order $3$ )

• A. $A$
• B. $B$
• C. $\frac{2I}{3}$
• D. $2I$

Answer 1

Answer: (C)

Given $A+B=2I$ ... (i)
and $A^{-1}+B^{-1}=3I$ ...(ii)
Multiplying by matrix $A$ in eq. (ii), we get,
$A{A}^{-1}+A{B}^{-1}=3AI$
$I+A{B}^{-1}=3A$
Now, multiplying by matrix $B$ , we get,
$IB+A{B}^{-1}B=3AB$
$\Rightarrow B+A=3AB$
$\Rightarrow 3AB=2I$ (from (i))
$\Rightarrow AB=\frac{2I}{3}$