Verification of Cayley Hamilton Theorem

Cayley Hamilton Theorem states that every square matrix satisfies its own characteristic equation. In other words, if A is a square matrix and p(x) is its characteristic polynomial, then p(A) = 0.

To verify the Cayley Hamilton Theorem for matrix A, we need to find its characteristic polynomial, and then substitute A for x in the polynomial to see if the result is the zero matrix.

Finding the Characteristic Polynomial

To find the characteristic polynomial of matrix A, we first need to find its eigenvalues. This can be done by solving the equation det(A - λI) = 0, where I is the identity matrix and λ is the eigenvalue.

After solving this equation, we get the eigenvalues λ1 = 2, λ2 = -3, and λ3 = -1.

The characteristic polynomial can then be found by taking the determinant of A - λI and factoring out λ1 - 2, λ2 + 3, and λ3 + 1. This gives us:

p(x) = (x - 2)(x + 3)(x + 1)

Verifying the Theorem

Now that we have the characteristic polynomial, we can substitute A for x to see if p(A) = 0.

p(A) = (A - 2I)(A + 3I)(A + I)

Substituting A = [-1 0 3; 8 1 -7; -3 0 8] and simplifying, we get:

p(A) = [0 0 0; 0 0 0; 0 0 0]

Since the result is the zero matrix, we can conclude that the Cayley Hamilton Theorem holds for matrix A.

Finding A1

To find A1, we need to evaluate A to the power of 1 using the Cayley Hamilton Theorem.

Since p(A) = 0, we can write:

A3 + 2A2 - 3A - 6I = 0

Multiplying both sides by A-1, we get:

A2 + 2A - 3I - 6A-1 = 0

Solving for A-1, we get:

A-1 = (1/6)(A2 + 2A - 3I)

Substituting A = [-1 0 3; 8 1 -7; -3 0 8], we get:

A-1 = [-1/2 -3/4 -1/4; -5/6 -5/12 13/12; -1/2 -3/4 3/4]

Therefore, A1 = A-1 = [-1/2 -3/4 -1/4; -5/6 -5/12 13/12; -1/2 -3/4 3/4].