Finding matrix power to 10 using characterstic polynomial.

Question

For these kind of problems we generally use diagonalization and similarity concept using eigen values and eigen vectors.

But the wording of question seems that we need to evaluate from characteristic polynomial without finding eigen vectors. In this suggest me an approach.

Answer 1

For this, we use Cayley-Hamilton Theorem, which states that if $P (x) $ is the characteristic polynom of the matrix $M $, then

$$P (M)=0$$

for example if $P (x)=x^3-1$, then $$M^3=I .$$

Answer 2

Using polynomial division ("long division"), write $$ x^{10} = q(x) \cdot p(x) + r(x) $$ where $p(x) = x^3 - 8 x^2 + 20x - 16$ is the characteristic polynomial, $q(x)$ and $r(x)$ are polynomials and the degree of $r$ is at most $2$. With that, conclude that $A^{10} = r(A)$.

With correct computation, you should find $$ r(x) = 2^8(1013 x^2 - 4032 x + 4016) $$