Matrix Inverse Calculator
Find the inverse of a square matrix. A matrix is invertible if and only if its determinant is nonzero. Uses Gauss-Jordan elimination.
Notes
What is the Matrix Inverse?
The inverse A⁻¹ of matrix A satisfies A·A⁻¹ = A⁻¹·A = I (the identity matrix). A matrix is invertible if and only if det(A) ≠ 0.
2×2 Direct Formula
Gauss-Jordan Method: Augmented Matrix
Form [A | I] and row-reduce until the left block becomes I. The right block is then A⁻¹.
Example for A = [[1,2],[3,4]]:
| 1 | 2 | 1 | 0 |
| 3 | 4 | 0 | 1 |
R2 → R2 − 3·R1, then eliminate upward, scale pivots:
| 1 | 0 | -2 | 1 |
| 0 | 1 | 3/2 | -1/2 |
A⁻¹ = [[-2, 1], [3/2, -1/2]]. Verify: A · A⁻¹ = I.
- Matrix Calculations Guide — Matrix Inverse — In-depth notes on Matrix Inverse with worked examples
Frequently Asked Questions
When does a matrix have no inverse?
When its determinant is zero. Such matrices are called singular. Their rows or columns are linearly dependent.
What is the formula for the 2×2 inverse?
For A = [[a,b],[c,d]], A⁻¹ = (1/(ad−bc)) · [[d,−b],[−c,a]].
How does Gauss-Jordan elimination find the inverse?
You augment A with the identity matrix [A|I] and row-reduce until the left side becomes I. The right side then equals A⁻¹.