Verify the Cayley-Hamilton Theorem for

The Cayley-Hamilton Theorem can be used to calculate powers and inverses of matrices. For example, if A is a 2 x 2 matrix with characteristic polynomial cA(λ) = λ2 + aλ + b, then A2 + aA + bl = 0, so
A2 = - aA - bl
and
A3 = AA2 = A(- aA - bl)
= - aA2 - bA
= - a(- aA - bl) - bA
= (a2 - b)A + ab I
It is easy to see that by continuing in this fashion we can express any positive power of A as a linear combination of I and A. From A2 + aA + bl = 0, we also obtain A(A + al ) = - bl, so

provided b ‰  0.

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