We use Cayley- Hamilton Theorem to compute*the power of the matrixthe inverse matrixdeterminant of the matrix*

The Cayley-Hamilton theorem is a fundamental theorem in linear algebra that states that every square matrix over a commutative ring (such as the real or complex numbers) satisfies its own characteristic equation.

1. Computing the power of a matrix using Cayley-Hamilton Theorem:

   Let's say we have a matrix A and its characteristic equation is given by:

   det(A - λI) = 0

   where I is the identity matrix and λ are the eigenvalues of the matrix A. The Cayley-Hamilton theorem states that if we replace λ by A in the characteristic equation, the resulting matrix equation holds:

   p(A) = 0

   where p(A) is the characteristic polynomial of A. This can be used to compute high powers of A. For example, if A is a 2x2 matrix with characteristic polynomial λ² - trace(A)λ + det(A), then A² can be expressed in terms of A and the identity matrix I:

   A² = trace(A)A - det(A)I

2. Computing the inverse of a matrix using Cayley-Hamilton Theorem:

   If A is an invertible matrix, then its characteristic polynomial has a non-zero constant term (which is det(A)), and we can write:

   A⁻¹ = -1/det(A) * adj(A)

   where adj(A) is the adjugate of A. This can be derived from the Cayley-Hamilton theorem by expressing A⁻¹ as a polynomial in A.

3. Computing the determinant of a matrix using Cayley-Hamilton Theorem:

   The determinant of a matrix A can be found directly from its characteristic polynomial. The constant term of the characteristic polynomial (when written in the form λ^n - ... + (-1)^n det(A)) is (-1)^n det(A), so:

   det(A) = (-1)^n * constant term

   where n is the order of the matrix.