Inverse Matrix Calculator
This inverse matrix calculator finds A⁻¹ for a 2×2, 3×3 or 4×4 matrix, shows the minors, the signs, the flip and the division in full, and proves the answer by multiplying it back out.
The quickest case first. For 4 7 / 2 6, the determinant is (4)(6) − (7)(2) = 10. Swap the two on the main diagonal and flip the sign of the other two to get 6 −7 / −2 4, then divide every entry by 10: the inverse is 3/5 −7/10 / −1/5 2/5. Multiply the original by that and the identity matrix comes back, which is what makes it an inverse rather than a plausible-looking grid of numbers.
An inverse is the matrix equivalent of a reciprocal. Just as 5 has 1/5 because the two multiply to 1, a square matrix A has A⁻¹ because the two multiply to the identity matrix — the grid with ones down the diagonal that leaves everything it touches unchanged.
Finding the Inverse of a 2×2 Matrix
Three moves, and they are worth memorising because every larger case is the same idea dressed up. Take a b / c d. Swap a and d, flip the signs of b and c, and divide the whole thing by ad − bc.
That divisor is the determinant, and everything depends on it. If it comes out as zero there is nothing to divide by, and no inverse exists at all — which is the subject of a section further down.
Finding the Inverse of a 3×3 Matrix
The shortcut runs out here, and the general method takes over. It has four stages, and the calculator shows each one as its own block.
Minors. For every position in the grid, cross out its own row and its own column and take the determinant of the 2×2 that is left. That number is the minor for that position.
Signs. Lay the sign grid over the top — a plus at the top left, then alternating in both directions, + − + / − + − / + − +. Multiply each minor by its sign and the minors become cofactors.
Flip. Turn the cofactor grid over its main diagonal so that rows become columns. What comes out is called the adjugate, or the adjoint in older books.
Divide. Divide every entry of the adjugate by the determinant, and that is A⁻¹.
Take 1 2 3 / 0 1 4 / 5 6 0, whose determinant works out to 1. The cofactors come to −24 20 −5 / 18 −15 4 / 5 −4 1, and after the flip the adjugate is −24 18 5 / 20 −15 −4 / −5 4 1. Dividing by 1 changes nothing, so that adjugate is the inverse — and because the division is invisible here, this example shows the flip very clearly.
The Step Everyone Forgets
Look again at those two grids. The cofactors and the adjugate hold the same nine numbers, but not in the same places, and an answer built on the un-flipped grid will look entirely reasonable and be wrong.
That is why this calculator gives the flip a block of its own rather than folding it into the arithmetic. It is also why the check at the end matters: a dropped flip survives every stage of the working and dies immediately when the product is worked out, because the result stops being the identity.
When a Matrix Has No Inverse
A square matrix is invertible, or non-singular, when its determinant is anything other than zero. When the determinant is zero the matrix is singular and has no inverse.
Take 1 2 / 2 4, where the second row is simply twice the first. The rows are linearly dependent, the determinant is exactly zero, and the transformation collapses the plane onto a line. Nothing that has been flattened can be unflattened, so there is no matrix that undoes it.
This is the place where rounding does real damage. Every entry of an answer is divided by the determinant, so a tool that carries decimals internally can divide by something like 0.0000000001 where the true value is zero, and hand back an inverse made of enormous numbers that is entirely fictional. Everything here is kept in fractions, so a zero determinant is a zero and the answer is a refusal rather than a fiction.
Inverses matter mostly because of what they solve. A system of equations written as AX = B has the solution X = A⁻¹B whenever A can be inverted, which is why a singular coefficient matrix and a system without a unique answer are the same fact wearing different clothes. If that is the problem in front of you, the system of equations calculator takes it directly.
How to Use This Inverse Matrix Calculator
The screen has three parts: the answer band at the top, the matrix you fill in below it, and the keypad underneath. You fill cells, the way you would write the matrix out by hand.
1. Set the size. Two rows of number buttons sit above the grid, one for rows and one for columns, from 1 to 4. They are separate on purpose — set them differently and the calculator explains why a rectangle has no inverse.
2. Fill in the cells. Tap a cell to select it, then use the keypad. A cell takes a whole number, a decimal such as 0.25, or a fraction such as 3/4, and anything left blank counts as zero.
3. Press =. A⁻¹ appears at the top in its brackets. If the matrix is singular the caption changes to SINGULAR and the line underneath says why.
4. Open the route. The button marked Show the whole route lays out every stage: the determinant, the 2×2 shortcut where it applies, the minors with each small matrix drawn separately, the sign grid, the flip, the division, and finally the product worked out to the identity.
Every key and control, in one place:
| rows / columns | Sets the size, 1 to 4, each one on its own |
| the cells | Tap to select; blank counts as zero; fractions and decimals both accepted |
| ← → | Move to the previous or next cell, wrapping round the whole grid |
| ↑ ↓ | Move up or down a row, for filling a column at a time |
| ± | Flips the sign of the cell you are in, so a minus cannot be typed twice by accident |
| ⁄ | The fraction bar. 3⁄4 stays a fraction all the way to the answer |
| . | Decimal point. 0.25 is read as one quarter, exactly |
| ⌫ | Deletes one character from the cell you are in |
| AC | Clears the whole matrix and starts again |
| = | Works out the inverse |
| My keyboard | Switches that cell to your own keyboard, for anything the pad does not carry |
| copy | The icon in the band copies the answer, row by row |
Nothing is shown until the answer has been multiplied back out and come back as the identity, and you can watch that happen in the last block. If you would like to run the product yourself, the matrix multiplication calculator will take both grids. Change anything after an answer is showing and the band dims and asks you to press = again, so what you are reading always belongs to what is in the cells.
Inverse Matrix FAQ
How do you find the inverse of a 2x2 matrix?
What is the adjugate of a matrix?
Why does a matrix need a non-zero determinant to have an inverse?
How do you check that an inverse is correct?
Can a non-square matrix have an inverse?
Why are the entries of an inverse usually fractions?
Minors, signs, flip, divide — and then multiply it back out to be sure. The first three stages are where the marks are lost and the last one is where you find out, which is why this calculator refuses to show an answer it has not already proved.