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Question

# (a) Give a counterexample to show that $$( A B ) ^ { - 1 } \neqA ^ { - 1 } B ^ { - 1 }$$ in general. (h) Under what conditions on A and B is $$( A B ) ^ { - 1 } =A ^ { - 1 } B ^ { - 1 } ?$$ Prove your assertion.

Solution

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(a)

First, we'll choose matrices $A$ and $B$, then show $(AB)^{-1}\ne A^{-1}B^{-1}$ directly. Matrices may seem arbitrary now but the only way to think of them yourself is trough practice. However, you should be able to check if the statement is true for given matrices.

$A=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\text{, }B=\begin{bmatrix} 1 & 0 \\ 0 & 0.5 \end{bmatrix}$

We know that, for a matrix $M=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ where $M$ is invertibile (in other words when $\det M=ad-bc\ne0$), we can find the inverse as

$M^{-1}=\dfrac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

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