Find the 30th power of a matrix

Question

Let $A$ be the following matrix: $$A=\left(\begin{array}{rrr} 1&0&0\\ 1&0&1\\ 0&1&0\\ \end{array}\right) $$ Find the matrix $A^{30}$.

Is this matrix diagonalizable? Is there any other method to find the higher power, when the matrix cannot be diagonalized?

Answer 1

Note that

$$A=\left(\begin{array}{rrr} 1&0&0\\ 1&0&1\\ 0&1&0\\ \end{array}\right) = \left(\begin{array}{rrr} 1&0&0\\ 1&1&0\\ 0&0&1\\ \end{array}\right) \left(\begin{array}{rrr} 1&0&0\\ 0&0&1\\ 0&1&0\\ \end{array}\right) $$ decomposes as two elementary matrices.

Answer 2

Hint:

Prove by induction that: $$ \left(\begin{array}{rrr} 1&0&0\\ 1&0&1\\ 0&1&0\\ \end{array}\right) ^{2n}= \left(\begin{array}{rrr} 1&0&0\\ n&1&0\\ n&0&1\\ \end{array}\right) $$