matrix - Verify Cayley-Hamilton Theorem

Consider the matrix:

A = {{-1, -4, -2}, {0, 1, 1}, {-6, -12, 2}}

Now, the characteristic polynomial is found:

p[λ_] = CharacteristicPolynomial[A, λ]

Now, how can I substitute the matrix A into the polynomial to verify that the answer is the zero matrix?

Answer

To verify that matrix is a zero of its characteristic polynomial, The Characteristic polynomial of the matrix is found, then evaluated for the matrix. The result should be the zero matrix.

Clear[x]
a = {{-1, -4, -2}, {0, 1, 1}, {-6, -12, 2}};

n = Length[a];
p = CharacteristicPolynomial[a, x];
(Sum[ Coefficient[p, x, i] MatrixPower[a, i], {i, 0, Exponent[p, x]}]) // MatrixForm

another way:

cl = CoefficientList[p, x];
Sum[MatrixPower[a, j - 1] cl[[j]], {j, 1, Length[cl]}]