# If A and B are invertible matrices,then which of the following is not correct?$\begin{array}{1 1}(A)\quad adjA=|A|.A^{-1} & (B)\quad det(A)^{-1}=[det(A)]^{-1}\\(C)\quad (AB)^{-1}=B^{-1}A^{-1} & (D)\quad (A+B)^{-1}=B^{-1}+A^{-1}\end{array}$

## 1 Answer

Toolbox:
• If $A^{-1}=\frac{1}{|A|}adj( A)$
• $A(adj A)=|A|I=(adj A)A$
• $(AB)^{-1}=B^{-1}A^{-1}$
Given A and B are invertible matrices.
Then $(A+B)^{-1}=B^{-}+A^{-1}$ is not correct.
For example let us consider
$A=\begin{bmatrix}1 & 2\\0 & 1\end{bmatrix}$ and $\begin{bmatrix}2 & 1\\2 & 3\end{bmatrix}$
|A|=1-0=1 and |B|=6-2=4
$adj \;of A=\begin{bmatrix}1 &-2\\0 & 1\end{bmatrix}$ and $adj B=\begin{bmatrix}3 & -1\\-2 & 2\end{bmatrix}$
$A^{-1}=\frac{1}{1}\begin{bmatrix}1 &-2\\0 & 1\end{bmatrix}$ and $B^{-1}=\frac{1}{4}\begin{bmatrix}3 & -1\\-2 & 2\end{bmatrix}$
$A+B=\begin{bmatrix}1 &2\\0 & 1\end{bmatrix}+\begin{bmatrix}2 & 1\\2 & 3\end{bmatrix}$
$\;\;\;\;\;\;=\begin{bmatrix}3 & 3\\2 & 4\end{bmatrix}$
|A+B|=12-6=6.
$adj (A+B)=\begin{bmatrix}4 & -3\\-2 & 3\end{bmatrix}$
$[A+B]^{-1}=\frac{1}{6}\begin{bmatrix}4 & -3\\-2 & 3\end{bmatrix}=\begin{bmatrix}4/6 & -3/6\\-2/6 & 3/6\end{bmatrix}=\begin{bmatrix}2/3 & -1/2\\-1/3 & 1/2\end{bmatrix}$-----LHS
$B^{-1}+A^{-1}=\frac{1}{4}\begin{bmatrix}3 & -1\\-2 & 2\end{bmatrix}+\begin{bmatrix}1 & -2\\0 & 1\end{bmatrix}=\begin{bmatrix}3/4+1 & -1/4-2\\-2/4+0 & 2/4+1\end{bmatrix}=\begin{bmatrix}7/4 & -9/4\\-1/2 & 6/4\end{bmatrix}$------RHS
Hence RHS$\neq$ LHS.
Hence D is the correct answer.
answered Mar 26, 2013

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