Finding Inverse of a Square Matrix using Cayley Hamilton Theorem in MATLAB

Last Updated : 18 Apr, 2022

Matrix is the set of numbers arranged in rows & columns in order to form a Rectangular array. Here, those numbers are called the entries or elements of that matrix. A Rectangular array of (m*n) numbers in the form of ‘m’ horizontal lines (rows) & ‘n’ vertical lines (called columns), is called a matrix of order m by n, which is written as m × n matrix.

Note: If m=n i.e., Number of Rows = Number of Columns in a Matrix ‘A’, then ‘A’ is said to be a Square Matrix of order ‘n’.

The inverse of a Square Matrix:

If ‘A’ is a non-singular (|A| ≠ 0) square matrix of order ‘n’, then there exists an n x n matrix A^-1, which is called the inverse matrix of ‘A’, which satisfies the following property:

AA^{-1 = A-1A = I}_ⁿ, where I_n is  the Identity matrix of order n

We can Find Inverse of a Square Matrix using various methods like:

Cayley-Hamilton Method
Gaussian Elimination
Newton’s Method
Eigen Decomposition Method

Cayley-Hamilton’s theorem:

In the Domain of Linear Algebra, According to Cayley-Hamilton’s theorem, every square matrix ‘A’ of order ‘n’ satisfies its own characteristic equation, which is given by:

|A-λIn| = 0 
Here 'In' is the Identity Matrix  
of Order 'n' (same as that of A's Order)
and 'λ' is some Real Constant.

Expanding the above characteristic equation of ‘A’, we get:

λⁿ + C₁λ^n-1 + C₂λ^n-2 + . . . + C_nI_n = 0, 
where C₁, C₂, . . . , C_n are Real Constants.

According to Cayley-Hamilton’s theorem, The above characteristic equation of ‘A’ is satisfied by itself, Hence we have:

Aⁿ + C₁A^n-1 + C₂A^n-2+ . . . + C_nI_n = 0 , 
where C₁, C₂, . . . , C_n are Real Constants.

Steps to Find Inverse of a Square Matrix using Cayley Hamilton theorem:

Step 1: For a Given Non-singular Square Matrix ‘A’ of order ‘n’, Find its Characteristic equation |A-λI_n| = 0.

Expand the Determinant such that it is reduced in the below format:

λ_n + C₁λ^n-1 + C₂λ^n-2 + . . . + C_nI_n = 0,
where C₁, C₂, . . . , C_n are Real Constants.

Step 2: As per Cayley-Hamilton’s theorem, the above Characteristic equation of ‘A’ is satisfied by itself, Hence:

Aⁿ + C₁A^n-1 + C₂A^n-2+ . . . + C_nI_n = 0

Step 3: Multiplying by A^-1 on Both Sides of the above Equation reduces it to:

A^n-1 + C₁A^n-2 + C₂A^n-3 + . . . C_n-1I_n + C_nA^-1 = 0

Step 4: Find A^-1 by simplifying and reordering the terms of the above equation, then A^-1 is:

A^{-1 = (-1/C}_{^{n)[ An-1 + C1An-2 + C2An-3 + . . . Cn-1In ]}}

MATLAB Functions used in the Below Code are:

disp(“txt”): This Method displays the message-‘txt’ to the User.
input(“txt”): This Method displays the message-‘txt’ and waits for the user to input a value and press the Return key.
poly(A): This method returns the n+1 coefficients of the Characteristic polynomial of the matrix ‘A’ of order ‘n’.
length(X): This method returns the number of elements of the vector ‘X’.
round(x): This Method rounds ‘x’ to its nearest integer.

Example:

Matlab

% MATLAB Implementation to find Inverse 
% of a Square Matrix using Cayley Hamilton theorem:
clear all  
clc        
disp("Finding Inverse of a Square Matrix  using
Cayley  Hamilton theorem in MATLAB | GeeksforGeeks")
A=input('Enter the Matrix A: ');
 
% To find Coefficients of Characteristic Equation of Matrix 'A'
cf=poly(A);  
 
% To find the Number of Coefficients in 
% the Characteristic Equation of Matrix 'A'
n=length(cf); 
 
% To find the Inverse of A
inverse = cf(1)*A^(n-2);
for i=2:n-1
inverse=inverse+cf(i)*A^(n-i-1);
end
 
% Checking whether |A|=0 or not
if  round(cf(n))==0   
    disp('Inverse of A does not exist as it is a singular matrix..')
else
   inverse=inverse/(-cf(n));
   disp('Inverse of A: ')
   disp(inverse)
end