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. $A=\left[\begin{array}{cc}3& 7\\ 2& 5\end{array}\right],B=\left[\begin{array}{cc}6& 8\\ 7& 9\end{array}\right]$

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

$\left|A+B\right|=9\times 14-9\times 15=-9\ne 0$
So, $\left(A+B\right)$ is invertible.
adj $\left(A+B\right)=\left[\begin{array}{cc}14& -15\\ -9& 9\end{array}\right]$

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

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.
$A^{-1}=\left[\begin{array}{cc}5& -7\\ -2& 3\end{array}\right]$

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

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

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

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