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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!

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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?

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The note on the inverse matrix has already been said by some users, so I'll make a note on notation.

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).

Maybe I'm saying this incompletely, so if I'm missing something, could somebody please fill in the blanks? Thank you.

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  • $\begingroup$ 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. $\endgroup$
    – Jeppe Stig Nielsen
    May 19, 2014 at 6:31

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