Look at the n^2 matrices:
I,A,A^2,...,A^(n^2).
Remember that the vector space M of matrices nxn is n^2, but you have n^2+1 matrices, so...?
Hello, a tad stuck on an algebra assignment.
The Cayley Hamilton theorem implies that, for any nxn matrix A over a field K, there is a polynomial p(x) with coefficents in K and degree n in x such that p(A)=0.
By considering the matrices A^i for ,0 less than or eqaul to (i) less than or equal to (n^2), prove, without using the cayley hamilton theorem, that there is a non-zero polynomial p(x) of degree at most (n^2) in x, such that p(A)=0.
Hints/pointers/answers will all be welcomed.
Thankyou
I will continue from my first comment.
...hence, n^2+1 matrices are linear dependence, in other words:
there exist t_0,t_1,...t_{n^2+1} scalars that not all of them is zero, which for them:
t_{n^2+1}A^{n^2+1}+...+t_1A+t_0I=0
or in other writing...
A is a root of f(x)=t_{n^2+1}x^{n^2+1}+...+t_1x+t_0I.