Show linear dependency of a ordered set of matrices

Question

If $A$ is a $2\times 2$ matrix and the ordered set $= \{A^0,A^1,...A^4\}$. Then prove that the ordered set is linearly dependent.

I know that linear dependency means $a_0A^0+a_1A^1+...a_4A^4=0$ for $a_i\ne0$. But how can i prove this when A is not given in the question?

Answer 1

You have $5$ matrices but the dimension is at most $4$.

Hence, they must be linearly dependent.

Answer 2

Hint: Think about matrices as $4$-dimensional vectors.

Answer 3

In fact, $\{A^0, A^1, A^2\}$ are already linearly dependent. The characteristic polynomial of $A$ has degree $2$, so $p_A(X) = a_0 + a_1 X + a_2 X^2$. By Cayley-Hamilton, $0=p_A(A) = a_0 A^0 + a_1 A^1 + a_2 A^2$. Of course, using dimensions is much easier.