MatrixLab

Matrix Inverse Calculator

The matrix that undoes this one.

2×2 matrix inverse

a₁₁

a₁₂

a₂₁

a₂₂

Inverse

[[0.6, -0.7], [-0.2, 0.4]]

det = 10

What you entered

Matrix: [[4, 7], [2, 6]]

1
Determinant
4×6 − 7×2= 10
2
Swap diagonal, negate off-diagonal (adjugate)
[[d, −b], [−c, a]]= [[6, -7], [-2, 4]]
3
Divide every entry by the determinant
adjugate ÷ 10= [[0.6, -0.7], [-0.2, 0.4]]

Result

Inverse: [[0.6, -0.7], [-0.2, 0.4]]

Multiplying this matrix by its inverse gives the identity matrix — the inverse "undoes" the transformation this matrix represents.

Inverse of a 2x2 matrix

The inverse swaps a and d, negates b and c, then divides everything by the determinant.

Matrix	Determinant	Inverse
[[1, 2], [3, 4]]	-2	[[-2.0000, 1.0000], [1.5000, -0.5000]]
[[2, 0], [0, 2]]	4	[[0.5000, 0.0000], [0.0000, 0.5000]]
[[4, 7], [2, 6]]	10	[[0.6000, -0.7000], [-0.2000, 0.4000]]
[[1, 2], [2, 4]]	0	None - singular
[[3, 8], [4, 6]]	-14	[[-0.4286, 0.5714], [0.2857, -0.2143]]

The second row is a pure scaling by 2, so its inverse is a scaling by one half - the clearest illustration of what an inverse does. The fourth has determinant zero and therefore no inverse at all, which is the matrix equivalent of dividing by zero. Multiplying any matrix by its inverse returns the identity, which is the quickest way to check an answer: [[1,2],[3,4]] times its inverse should give [[1,0],[0,1]] up to rounding.

Swap, negate, divide

For a 2×2 matrix, the inverse is found by swapping the main diagonal, negating the other two entries (the adjugate), then dividing every entry by the determinant — no row-reduction needed at this size.

What the inverse is for

Multiplying a matrix by its inverse gives the identity matrix — this is exactly how systems of linear equations (Ax = b) are solved directly as x = A⁻¹b, and why a zero determinant means no inverse, and no unique solution, exists.

Frequently asked questions

Find the inverse of [[4, 7], [2, 6]]

det = 4(6) − 7(2) = 24 − 14 = 10. Adjugate: swap diagonal [[6, −7], [−2, 4]]. Divide each by 10: [[0.6, −0.7], [−0.2, 0.4]]. Verify: multiply original × inverse and you get the identity matrix [[1, 0], [0, 1]].

My matrix [[2, 4], [1, 2]] has no inverse — why?

Its determinant is 2(2) − 4(1) = 0. A zero determinant means the matrix is singular. Geometrically, this matrix collapses 2D space onto a line — the transformation loses information and can't be reversed. The second row is exactly half the first, so they're linearly dependent.

How is the matrix inverse used to solve systems of equations?

For Ax = b, multiply both sides by A⁻¹: x = A⁻¹b. Example: 4x + 7y = 5, 2x + 6y = 3. Matrix A = [[4,7],[2,6]], b = [5,3]. Using the inverse above: x = [[0.6,−0.7],[−0.2,0.4]] × [5,3] = [0.9, 0.2]. So x = 0.9, y = 0.2.

What's the relationship between the inverse and the determinant?

Every element in the 2×2 inverse is divided by the determinant. A larger determinant means smaller inverse elements (less 'amplification'), and det = 0 makes division impossible. The matrix determinant calculator can check invertibility before you attempt the full inverse computation.

Does every square matrix have an inverse?

No — only matrices with a non-zero determinant. These are called 'invertible' or 'non-singular.' Roughly half of random integer matrices are singular. In practice, near-zero determinants also cause problems because tiny rounding errors get amplified hugely.