Understanding Inverse Matrices with Powers

[courtrigrad]

If $$A = [a_{ij}]^{n\times n}$$ is invertible, show that $$(A^{2})^{-1} = (A^{-1})^{2}$$ and $$(A^{3})^{-1} = (A^{-1})^{3}$$

So basicaly we have a square matrix with elements $$a_{ij}$$. This looks slightly familar to $$(A^{T})^{-1} = (A^{-1})^{T}$$. Are $$A^{2}$$ and $$A^{3}$$ meant to be the elements of the matrix raised to those respective powers? Or does it mean that the matrix is $$2\times 2$$ or $$3\times 3$$?

 

[quasar987]

The matrix is nxn. A^2 means AA.

As 90% of linear algebra proofs, these problems are solvable in 2-3 lines. If you really don't find it, I can start you and you will find it imidiately. Laying the first equation is the hardest.

 

[courtrigrad]

thanks. I got it, just wasn't clear about the notation.