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