About the Matrix Inverse Calculator
This matrix inverse calculator finds A⁻¹ for a square matrix from 2×2 up to 6×6 using Gauss–Jordan elimination on the augmented matrix [A | I]. Every row operation — swapping rows, scaling a row to make a leading 1, and subtracting multiples of the pivot row — is listed in order, together with the augmented matrix after each step, so you can follow or check the working by hand.
All arithmetic is done with exact fractions, so the inverse of a matrix with whole-number entries is shown with exact entries such as −7/10 rather than a rounded decimal (a decimal version is shown too). Fractions and decimals are accepted as input.
Students use it to check homework, and engineers and analysts use it to solve systems of equations x = A⁻¹b or to undo a linear transformation. A matrix has an inverse only when it is square and its determinant is non-zero; if elimination finds a column with no usable pivot, the calculator reports that the matrix is singular.
How to use the matrix inverse calculator
- 1Type matrix A with one row per line and spaces or commas between entries.
- 2Read the exact inverse (fractions) and its decimal form.
- 3Follow the row operations table to see how [A | I] becomes [I | A⁻¹].
- 4If the matrix is singular, check whether one row is a combination of the others.
Formula and method
Gauss–Jordan elimination writes the identity matrix next to A and applies the same elementary row operations to both halves until the left half becomes the identity. Because each row operation is equivalent to multiplying on the left by an invertible matrix, the product of all of them is A⁻¹ — and that is exactly what has been built up on the right.
Column by column, the calculator picks a non-zero pivot (swapping rows if needed), divides the pivot row to make the pivot 1, then subtracts multiples of the pivot row from every other row to clear the column. If some column has no non-zero pivot, A is singular (det A = 0) and no inverse exists.
- A
- Square input matrix
- I
- Identity matrix of the same size
- A⁻¹
- Inverse: A·A⁻¹ = A⁻¹·A = I
- Rᵢ
- Row i of the augmented matrix
Worked examples
3×3 inverse with integer entries
det(A) = −1, so the inverse has integer entries. Gauss–Jordan elimination on [A | I] ends with the identity on the left and A⁻¹ on the right; multiplying A by this matrix gives the identity.
2×2 inverse with fractions
det = 4·6 − 7·2 = 10. Swap the diagonal entries, negate the off-diagonal ones and divide by 10: [[6, −7], [−2, 4]] ÷ 10 = [[3/5, −7/10], [−1/5, 2/5]].
Singular matrix
Row 3 equals 2 × row 2 − row 1, so the rows are dependent, det = 0 and elimination runs out of pivots in the last column. The matrix has no inverse.
Frequently asked questions
How do you find the inverse of a matrix?+
Write [A | I], use row operations to turn the left half into the identity, and read A⁻¹ from the right half. For a 2×2 matrix there is a shortcut: swap a and d, negate b and c, and divide by ad − bc.
When does a matrix not have an inverse?+
A matrix has no inverse if it is not square or its determinant is zero. Equivalently, its rows or columns are linearly dependent, or Ax = 0 has a non-zero solution.
How do I use the inverse to solve a system of equations?+
Write the system as Ax = b. If A is invertible, x = A⁻¹b. In practice elimination on [A | b] is faster, but the inverse is handy when you need to solve for many different right-hand sides b.
Is the inverse of a product the product of the inverses?+
Yes, but in reverse order: (AB)⁻¹ = B⁻¹A⁻¹. Also (Aᵀ)⁻¹ = (A⁻¹)ᵀ and (kA)⁻¹ = (1/k)A⁻¹ for any non-zero number k.
Why are the answers shown as fractions?+
The calculator works in exact rational arithmetic, so entries like 1/3 are not rounded to 0.333333. Exact fractions make it easy to verify A × A⁻¹ = I by hand; the decimal version is shown underneath.