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

Answer 1

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.

\[\text{We also know that } A^{- 1} + B^{- 1}\text{ is not the same as }\left( A + B \right)^{- 1} .\]