Linear Algebra Intermediate

Matrix Inverse (A⁻¹)

Find A⁻¹ using the adjugate formula for 2×2 matrices, or Gauss-Jordan row reduction for 3×3 matrices. Every row operation on the augmented matrix [A | I] is shown step by step.

Subject: Linear Algebra
Input: 2×2 or 3×3 matrix
Output: A⁻¹ + row reduction steps
Live
Matrix Input
Result
Enter matrix values and click Find Inverse

Step-by-Step Work

What the Inverse Means
A · A⁻¹ = A⁻¹ · A = I

The inverse of a matrix A "undoes" the linear transformation A applies. It only exists when det(A) ≠ 0 (the matrix is invertible / non-singular).

For 2×2: A⁻¹ = (1/det)·adj(A). For larger matrices, augment [A | I] and row-reduce — when the left side becomes I, the right side is A⁻¹.

Exam tip — check your answer by multiplying A · A⁻¹ and verifying you get the identity matrix.
Key Properties
  • (AB)⁻¹ = B⁻¹A⁻¹
  • (Aáµ€)⁻¹ = (A⁻¹)áµ€
  • (A⁻¹)⁻¹ = A
  • det(A⁻¹) = 1 / det(A)
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Put It Into Practice

The calculator shows the steps — a quiz proves you can do them on your own.

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