Inverse Matrix Calculator

What Is a Matrix Inverse?

The inverse of a square matrix A, written A⁻¹, is the unique matrix such that:

where I is the identity matrix. Not every matrix has an inverse — only those with a non-zero determinant.

Inverse of a 2×2 Matrix

For A = [[a,b],[c,d]] with det(A) = ad - bc ≠ 0:

Example: A = [[2,1],[5,3]], det = 2×3 - 1×5 = 1

A⁻¹ = [[3,-1],[-5,2]]

Verify: A × A⁻¹ = [[2×3+1×(-5), 2×(-1)+1×2],[5×3+3×(-5), 5×(-1)+3×2]] = [[1,0],[0,1]] = I ✓

Inverse of a 3×3 Matrix

The general formula uses the adjugate (transpose of cofactor matrix):

The calculator applies this formula automatically using cofactor expansion.

Properties of the Inverse

• (A⁻¹)⁻¹ = A
• (AB)⁻¹ = B⁻¹ × A⁻¹ (order reversed)
• (A^T)⁻¹ = (A⁻¹)^T
• det(A⁻¹) = 1/det(A)
• (kA)⁻¹ = (1/k) × A⁻¹ for scalar k ≠ 0

Real-World Applications

• Solving linear systems: Ax = b has unique solution x = A⁻¹b when A is invertible.
• Computer graphics: The inverse transformation undoes rotations, scalings, and translations.
• Cryptography: Hill cipher decryption uses the modular inverse of the key matrix.
• Statistics: Linear regression computes β = (X^T X)⁻¹ X^T y to find best-fit coefficients.

Frequently Asked Questions

When does a matrix have an inverse?

A square matrix is invertible if and only if det(A) ≠ 0. A singular matrix (det = 0) has no inverse.

What is A × A⁻¹?

A × A⁻¹ = A⁻¹ × A = I (the identity matrix). This is the defining property of the inverse.

How do you find the inverse of a 2×2 matrix?

For [[a,b],[c,d]]: swap a and d, negate b and c, divide everything by det = ad-bc.

What is the inverse of a matrix product?

(AB)⁻¹ = B⁻¹ × A⁻¹ — note the reversed order. Analogous to the reverse rule for transposes.