Exponentiation of non-diagonalizable matrix $M=\begin{bmatrix} 12 & 1\\ -9 & 6 \end{bmatrix}$

Question

Let

$$M=\begin{bmatrix} 12 & 1\\ -9 & 6 \end{bmatrix}$$

Find an expression for $M^n$.

I am not sure how to do it since $M$ is non-diagonalizable. Thank you for any help!

See case 2 of https://math.stackexchange.com/a/1538095/265466. — amd, Apr 10 '20 at 22:49

score 9 · Accepted Answer · answered Apr 10 '20 at 20:18

9

Let $X=M-9I$. Then $X^2=0$ and you may evaluate the binomial expansion of $M^n=(9I+X)^n$ easily.

answered Apr 10 '20 at 20:18

user1551

is there a name for this technique? – Tanner Apr 10 '20 at 20:29
@Tanner $X$ is called a nilpotent matrix in linear algebra. – user1551 Apr 10 '20 at 20:36

1 Answers1