The Inverse Of A Matrix Calculator

The inverse of a matrix is the matrix that undoes it, and for the two-by-two case the recipe is shorter than the explanation.

Top-left entry (a)The entry in row 1, column 1.Top-right entry (b)The entry in row 1, column 2.Bottom-left entry (c)The entry in row 2, column 1.Bottom-right entry (d)The entry in row 2, column 2.

Determinant

Inverse top-left: 0.6
Inverse top-right: -0.7
Inverse bottom-left: -0.2
Inverse bottom-right: 0.4

Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.

The formula this calculator uses

A inverse = (1 / (ad - bc)) x [ d, -b; -c, a ]

det: Determinant
a, b, c, d: Matrix entries
A inverse: Inverse matrix

How to check the result by hand

1
Write the matrix in a, b, c, d order
Label the four entries as a top-left, b top-right, c bottom-left, d bottom-right. Every later step refers to these letters.
2
Compute the determinant
Multiply a by d, multiply b by c, and subtract the second from the first. If the result is zero, stop because the matrix has no inverse.
3
Swap and negate
The new top-left is d, the new bottom-right is a, the new top-right is minus b and the new bottom-left is minus c.
4
Divide by the determinant and verify
Divide each of the four entries by the determinant, then multiply the original matrix by your answer to confirm you get the identity.

For worked examples, common mistakes and the limits of this formula, read the full How To Calculate The Inverse Of A Matrix page.