13.15: Matrix Inverse

A square matrix A is invertible if there is another matrix A^-1 such that:

A * A-1 = I   and A-1 * A = I

where I is the identity matrix. In MATLAB, the inverse can be found with inv, but you should use it carefully.

Example \(\PageIndex{1}\)

Inverse of a matrix.

A = [1 2 4; 4 5 6; 3 8 9];

Ainv = inv(A)

% Check the result

result = A * Ainv

Solution

Ainv =

-0.103448 0.482759 -0.275862
-0.620690 -0.103448 0.344828
0.586207 -0.068966 -0.103448

result =

1.0000 0 0
0 1.0000 0
0 -0.0000 1.0000

The result of A*Ainv should be close to the identity matrix. It may not look exactly perfect because computers use finite precision arithmetic.

Singular Matrix

A singular matrix is a square matrix that does not have an inverse.

Equivalently, a square matrix AAA is singular if its determinant is zero:

\(det⁡(A)=0\)

A singular matrix is important because if AAA is singular, then a system written as:

A*x = b

does not have a unique solution. It may have no solution or infinitely many solutions.

Example \(\PageIndex{1}\)

Singular matrix.

B = [1 2; 5 10];

Ainv = inv(B)

Solution

warning: matrix singular to machine precision
Ainv =

Inf Inf
Inf Inf

The matrix B is singular because its rows are dependent. MATLAB will warn you that the matrix is singular or close to singular.