Question

If $A=\left[\begin{array}{cc}3& 4\\ 2& 4\end{array}\right],B=\left[\begin{array}{cc}-2& -2\\ 0& -1\end{array}\right],\mathrm{then}{\left(A+B\right)}^{-1}=$

(a) is a skew-symmetric matrix
(b) A⁻¹ + B⁻¹
(c) does not exist
(d) none of these

Answer 1

(d) none of these

$$\begin{gathered} \mathrm{We}\mathrm{have} \\ \left(A+B\right)=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right] \\ \therefore \left|A+B\right|=-1\ne 0 \\ \mathrm{Thus},{\left(A+B\right)}^{-1}\mathrm{exists}. \\ \mathrm{Now}, \\ {\left(A+B\right)}^T=\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right] \\ \mathrm{Here}, \\ {\left(A+B\right)}^T\ne -\left(A+B\right) \\ \mathrm{Hence},\mathrm{it}\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{skew}\mathrm{symmetric}\mathrm{matrix}. \\ \mathrm{We}\mathrm{also}\mathrm{know}\mathrm{that}{A}^{-1}+B^{-1}\mathrm{is}\mathrm{not}\mathrm{the}\mathrm{same}\mathrm{as}{\left(A+B\right)}^{-1}. \end{gathered}$$