Matrix Inverse Calculator

Find the inverse of a 2x2 or 3x3 matrix with detailed step-by-step solutions.

Enter Matrix

Matrix A

Result

A-1 =
Determinant 0
Invertible Yes

Step-by-Step Solution

A-1 = (1/det(A)) × adj(A)

What is a Matrix Inverse?

The inverse of a square matrix A, denoted A-1, is the matrix such that A × A-1 = A-1 × A = I, where I is the identity matrix. Not every matrix has an inverse; a matrix is invertible (non-singular) only when its determinant is non-zero.

Methods for Finding the Inverse

2x2 Formula

For a 2x2 matrix [[a,b],[c,d]], the inverse is computed using the adjugate formula.

A-1 = (1/(ad-bc)) [[d,-b],[-c,a]]

3x3 Cofactor Method

Compute the matrix of cofactors, transpose it to get the adjugate, then divide by the determinant.

A-1 = (1/det(A)) × adj(A)

Row Reduction

Augment A with the identity and row reduce to reduced row echelon form.

[A | I] → [I | A-1]

Singular Matrices

If det(A) = 0, the matrix is singular and has no inverse.

det(A) = 0 ⇒ no inverse

Properties of Matrix Inverse

  • (A-1)-1 = A
  • (AB)-1 = B-1A-1
  • (AT)-1 = (A-1)T
  • det(A-1) = 1/det(A)
  • (kA)-1 = (1/k)A-1 for scalar k

Applications

Matrix inversion is used to solve systems of linear equations (x = A-1b), in computer graphics for undoing transformations, in statistics for regression analysis, in control theory, and in cryptography. Although direct inversion is not always the most numerically stable method, understanding it is fundamental to linear algebra.