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.