Evaluate a matrix with a negative power

I am having problem with how to calculate a matrices that are raised to negative powers. I can manage the adding, multiplication etc, but I am stuck here.

The matrix in question is $A=\begin{bmatrix}5&-2\\10&-4\end{bmatrix}$. It is a $2\times 2$ matrix. I need to find what this matrix is raised to the power $-1$.

I preferably don't just want an answer, but a summary of how to do these as I have numerous others that I need to do. Any help will be much appreciated!

• – user7530 Dec 30 '13 at 23:58

You should know that if $A$ is a square matrix, then $A^{-1}$ denotes its inverse matrix if it exists. For the particular matrix you mentioned, the inverse does not exist. Can you see why?
Sometimes one finds matrices raised to powers like $-3/2$, $1/3$ and so on (I have found these, in particular). What is meant with this notation (at least according to what some teachers told me) is the following: that power ($-3/2$, $1/3$) of the matrix in diagonal form (i.e., that power of every eigenvalue of the matrix).
• That method works (only) for some matrices. In the above case, the entries on the diagonal of the diagonalized matrix are $0$ and $1$. Then the problem becomes to calculate $0^{-1}$ which is not defined. Other matrices do not even allow a "diagonal form", so you can't use the diagonal form to calculate (or define) functions of matrices for those. There exist non-diagonalizable matrices $B$ that are invertible, so $B^{-1}$ exists. Non-diagonalizable matrices can also be singular. – Jeppe Stig Nielsen May 19 '14 at 6:31