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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 $(A+B)^{-1}$.

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