Matrix Inverse Calculator
Free matrix inverse calculator. Invert a 2x2, 3x3 or larger square matrix with Gauss-Jordan elimination, get exact fractions, and see the A·A⁻¹ = I verification.
Invert any square matrix by Gauss-Jordan elimination. Results come back as exact fractions, and the product A·A⁻¹ is shown so you can confirm it really is the identity.
Facts about inverses
- (AB)⁻¹ = B⁻¹A⁻¹ — like transposes, inverses reverse the order of a product.
- det(A⁻¹) = 1/det(A) — which is exactly why a zero determinant blocks inversion.
- (A⁻¹)⁻¹ = A — inverting twice gets you back where you started.
- Prefer solving over inverting. To solve Ax = b, use elimination or an LU factorization rather than computing A⁻¹ — it is faster and numerically safer.
Frequently asked questions
How do you find the inverse of a matrix? ▾
Write the matrix next to an identity matrix of the same size to form [A | I], then row reduce until the left half becomes the identity. Whatever the right half has turned into is A⁻¹.
Which matrices have an inverse? ▾
Only square matrices with a non-zero determinant. If the determinant is zero the matrix is singular: its rows are linearly dependent and no inverse exists.
What is the formula for a 2x2 inverse? ▾
For rows (a, b) and (c, d), the inverse is 1/(ad − bc) times the matrix with rows (d, −b) and (−c, a). Swap the diagonal entries, negate the off-diagonal ones, and divide by the determinant.
How do I check an inverse is right? ▾
Multiply A by your answer. The product must be the identity matrix. This calculator performs that check and displays the result.
About this Calculator
Free matrix inverse calculator. Invert a 2x2, 3x3 or larger square matrix with Gauss-Jordan elimination, get exact fractions, and see the A·A⁻¹ = I verification.