Question

If A and B are invertible matrices, then which of the following is not correct?

• A. adj $A$ $=\left|A\right|$. $A^{-1}$
• B. $det\left(A\right)^{-1}=\left[det\left(A\right)\right]^{-1}$
• C. $\left(AB\right)^{-1}=B^{-1}A^{-1}$
• D. $\left(A+B\right)^{-1}=B^{-1}+A^{-1}$

Answer 1

Answer: (D)

$\left(A+B\right)^{-1} \ne B^{-1}+A^{-1}$
e. g. $A=\left[\begin{matrix}3&7\\ 2&5\end{matrix}\right], B=\left[\begin{matrix}6&8\\ 7&9\end{matrix}\right]$

$A + B =\left[\begin{matrix}9&15\\ 9&14\end{matrix}\right]$

$\left|A+B\right|=9\times14-9\times15=-9\ne0$
So, $\left(A + B\right)$ is invertible.
adj $\left(A + B\right) =\left[\begin{matrix}14&-15\\ -9&9\end{matrix}\right]$

$\left(A+B\right)^{-1}=\frac{-1}{9}\left[\begin{matrix}14&-15\\ -9&9\end{matrix}\right]=\left[\begin{matrix}\frac{-14}{9}&\frac{15}{9}\\ 1&-1\end{matrix}\right]$

Now, $|A| = 3 \times 5 - 2 \times 7 = 15- 14 = 1 \ne 0$
and $|B| = 6 \times 9 - 7 \times 8 = 54 - 56 = -2 \ne 0$
So, $A$ and $B$ both are invertible.
$A^{-1}=\left[\begin{matrix}5&-7\\ -2&3\end{matrix}\right]$

and $B^{-1}=\frac{-1}{2}\left[\begin{matrix}9&-8\\ -7&6\end{matrix}\right]\quad\Rightarrow \quad B^{-1}=\left[\begin{matrix}\frac{-9}{2}&4\\ \frac{7}{2}&-3\end{matrix}\right]$

Now, $B^{-1}+A^{-1}=\left[\begin{matrix}-\frac{9}{2}&4\\ \frac{7}{2}&-3\end{matrix}\right]+\left[\begin{matrix}5&-7\\ -2&3\end{matrix}\right]$

$=\left[\begin{matrix}\frac{1}{2}&-3\\ \frac{3}{2}&0\end{matrix}\right]\ne\left[\begin{matrix}-\frac{14}{9}&\frac{15}{9}\\ 1&-1\end{matrix}\right]=\left(A+B\right)^{-1}$

Hence, $\left(A+B\right)^{-1}\ne B^{-1}+A^{-1}$