Properties of Matrix Product I am studying for my exam next week and the teacher has posted previous exams online.
I have the following question, given:
$A = \begin{bmatrix}4&-2&2\\2&4&-4\\1&1&0\end{bmatrix}$
$u = \begin{bmatrix}1\\3\\2\end{bmatrix}$
Find $A^5*u$ without any calculations...
HINT: Properties of the Matrix Product

Note that
$$A  u = \begin{pmatrix}4 & -2 & 2\\ 2 & 4 & -4\\1 & 1 & 0\end{pmatrix}\begin{pmatrix}1\\3\\2\end{pmatrix} = \begin{pmatrix}2 \\ 6 \\ 4\end{pmatrix} = 2 \begin{pmatrix}1 \\ 3 \\ 2\end{pmatrix} = 2 u$$
Then for any $n>0$ 
$$A^n u = A^{n-1} (A u)= 2^n u$$
