find the inverse using cayley hamilton theorem. use a 3x3 matrix and find the characteristic polynomial in matlab

To find the inverse of a 3x3 matrix using Cayley-Hamilton theorem in MATLAB, we need to follow these steps:

1. Define the 3x3 matrix A.
2. Calculate the characteristic polynomial of matrix A.
3. Evaluate the characteristic polynomial at matrix A.
4. Use the Cayley-Hamilton theorem which states that for a square matrix A, A satisfies its own characteristic polynomial, and therefore A^n can be expressed as a linear combination of powers of A up to n-1.
5. Compute the inverse of the matrix A.

Here's MATLAB code to find the inverse of a 3x3 matrix A using Cayley-Hamilton theorem:

main.m
% Define the 3x3 matrix A
A = [1, 2, 3;
     4, 5, 6;
     7, 8, 10];

% Calculate the characteristic polynomial of matrix A
syms lambda;
char_poly = det(A - lambda*eye(3));

% Evaluate the characteristic polynomial at matrix A
char_poly_A = double(subs(char_poly, lambda, A));

% Use Cayley-Hamilton theorem to find the inverse
% The inverse of A can be calculated as: A^2 - trace(A)*A + det(A)*I
inv_A = A^2 - trace(A)*A + det(A)*eye(3);

% Display the inverse of A
disp(inv_A);
481 chars
19 lines

In this code, we have defined a 3x3 matrix A and followed the steps mentioned above to find its inverse using the Cayley-Hamilton theorem.