Question:
If $A=\left[\begin{array}{ll}3 & 4 \\ 2 & 4\end{array}\right], B=\left[\begin{array}{cc}-2 & -2 \\ 0 & -1\end{array}\right]$, then $(A+B)^{-1}=$
(a) is a skew-symmetric matrix
(b) $A^{-1}+B^{-1}$
(c) does not exist
(d) none of these
Solution:
(d) none of these
We have
$(A+B)=\left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]$
$\therefore|A+B|=-1 \neq 0$
Thus, $(A+B)^{-1}$ exists.
Now,
$(A+B)^{T}=\left[\begin{array}{ll}1 & 2 \\ 2 & 3\end{array}\right]$
Here,
$(A+B)^{T} \neq-(A+B)$
Hence, it is not a skew symmetric matrix.
We also know that $A^{-1}+B^{-1}$ is not the same as $(A+B)^{-1}$.