Inverse Matrix Calculator
This inverse matrix calculator finds the inverse of a 2×2, 3×3 or 4×4 matrix with exact fractions. It shows the ad − bc formula for 2×2 matrices and every Gauss–Jordan row operation for larger ones, then checks that A × A⁻¹ is the identity matrix.
Result
Enter values and press Calculate to see the result.
Formula
A 2×2 inverse: swap a and d, change the signs of b and c, divide by the determinant.
Gauss–Jordan elimination: row-reduce A to the identity, doing the same row operations to I.
How to use this calculator
- Choose the size of the matrix: 2 × 2, 3 × 3 or 4 × 4.
- Fill in the grid row by row. Fractions such as 3/4 and decimals are kept exact.
- Press Calculate. You get A⁻¹ — the multiplicative inverse matrix, so A × A⁻¹ = I — with the determinant and every row operation.
Worked examples
Easy: a 2 × 2 matrix with determinant 1
det = 2 · 3 − 1 · 5 = 1, which is not 0, so the inverse exists.
Swap 2 and 3 and change the signs of 1 and 5: [[3, −1], [−5, 2]].
Divide by the determinant (1): the inverse is [[3, −1], [−5, 2]].
Medium: a 2 × 2 inverse with fractions
det = 4 · 6 − 7 · 2 = 10.
A⁻¹ = 1/10 · [[6, −7], [−2, 4]] = [[3/5, −7/10], [−1/5, 2/5]].
Check: 4 · 3/5 + 7 · (−1/5) = 12/5 − 7/5 = 1.
Hard: a 3 × 3 inverse by Gauss–Jordan
det = 1(0 − 24) − 2(0 − 20) + 3(0 − 5) = −24 + 40 − 15 = 1.
Row-reduce [A | I]: R₃ → R₃ − 5R₁, then clear column 2 with R₂, then column 3.
The right-hand half becomes A⁻¹ = [[−24, 18, 5], [20, −15, −4], [−5, 4, 1]].
Common mistakes
- Trying to invert a matrix with determinant 0 — it has no inverse (it is singular).
- For a 2×2, swapping b and c instead of a and d.
- Doing a row operation to the left half only: every operation applies to the whole row of [A | I].
- Dividing only some entries by the determinant.
Frequently asked questions
How do you find the inverse of a 3×3 matrix?
Write [A | I] and use row operations — swap rows, scale a row, add a multiple of one row to another — until the left half is the identity. The right half is then A⁻¹.
When does a matrix have no inverse?
When its determinant is 0. Such a matrix is called singular; its rows are linearly dependent. The calculator tells you instead of giving an answer.
What is the multiplicative inverse of a matrix?
The matrix A⁻¹ with A × A⁻¹ = A⁻¹ × A = I, the identity matrix — just as 1/5 is the multiplicative inverse of 5.
How is the answer checked?
The calculator multiplies A by the inverse it found, and the product must be exactly the identity matrix.