Question

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

• A. is a skew-symmetric matrix
• B. $A^{-1}+B^{-1}$
• C. does not exist
• D. none of these

Answer 1

The correct option is D none of these

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

$|A+B|=3-4=-1\ne 0$

Hence, inverse of $(A+B)$ exists.
Now, $adj(A+B)=C^T={\left[\begin{array}{cc}3& -2\\ -2& 1\end{array}\right]}^T$

$\Rightarrow adj(A+B)=\left[\begin{array}{cc}3& -2\\ -2& 1\end{array}\right]$

$\Rightarrow (A+B{)}^{-1}=\left[\begin{array}{cc}-3& 2\\ 2& -1\end{array}\right]$

Not a symmetric matrix.
Now, we will find $A^{-1},B^{-1}$

Here, $|A|=4,|B|=2$

$adjA=C^T={\left[\begin{array}{cc}4& -2\\ -4& 3\end{array}\right]}^T$

$\Rightarrow adjA=\left[\begin{array}{cc}4& -4\\ -2& 3\end{array}\right]$

$\Rightarrow A^{-1}=\left[\begin{array}{cc}1& -1\\ -1/2& 3/4\end{array}\right]$
$adjB=C^T={\left[\begin{array}{cc}-1& 0\\ 2& -2\end{array}\right]}^T$

$\Rightarrow adjB=\left[\begin{array}{cc}-1& 2\\ 0& -2\end{array}\right]$

$\Rightarrow B^{-1}=\left[\begin{array}{cc}-1/2& 1\\ 0& -1\end{array}\right]$
Hence, $A^{-1}+B^{-1}=\left[\begin{array}{cc}1& -1\\ -1/2& 3/4\end{array}\right]+\left[\begin{array}{cc}-1/2& 1\\ 0& -1\end{array}\right]=\left[\begin{array}{cc}1/2& 0\\ -1/2& -1/4\end{array}\right]$
Therefore, $(A+B{)}^{-1}\ne A^{-1}+B^{-1}$

Hence, option 'D' is correct.