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I had a matrices exam last week and I wrote $A^{-2}$ to refer to $(A^{-1})^2$ (A being an invertible matrix). Initially, I was given the question wrong, but I told the professor that I saw in a book that $(A^{-1})^n = A^{-n} = (A^n)^{-1}$ and gave me the point.

He said that $A^{-n}$ isn't defined for matrices, which I may believe since I haven't seen it in this forum yet, but I would find having to write $(A^{-1})^n$ or $(A^n)^{-1}$ very annoying just for the fact that it isn't defined.

So, is it (defined)?

P.D.: The book is Linear Algebra by Paul Dawkins 1.

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1 Answer

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Yes, $A^{-n}$ is defined to be $(A^{-1})^n$ when the matrix is invertible.

For example for $$A=\begin {bmatrix}3&2\\4&5\end {bmatrix}$$

We have $$ A^{-1} = \begin {bmatrix}5/7&-2/7\\-4/7&3/7\end {bmatrix}$$

Thus $$ A^{-2} =\begin {bmatrix}5/7&-2/7\\-4/7&3/7\end {bmatrix}^2= \begin {bmatrix}33/49&-16/49\\-32/49&17/49\end {bmatrix}$$

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