Matrix Inverse Calculator
Find the inverse of a 2×2 or 3×3 matrix using the adjugate over the determinant, with a check for singular matrices that have no inverse.
How it works
A⁻¹ = adj(A) ÷ det(A) · 2×2: A⁻¹ = 1/(ad − bc) · [d, −b; −c, a]
The inverse of a matrix undoes it: A × A⁻¹ equals the identity, just as multiplying a number by its reciprocal gives one. It is built from the adjugate — the transpose of the cofactor matrix — divided by the determinant. Because you divide by the determinant, a matrix with determinant zero (singular) has no inverse at all, the same way you cannot divide by zero. The inverse is what lets you solve A·x = b as x = A⁻¹·b.
Worked example
For A = [4, 7; 2, 6], the determinant is 4×6 − 7×2 = 10. Swap the diagonal, negate the off-diagonal, and divide by 10: A⁻¹ = 1/10 × [6, −7; −2, 4] = [0.6, −0.7; −0.2, 0.4]. Check: A × A⁻¹ gives [1, 0; 0, 1], the identity.
Sources: Invertible matrix — Wikipedia
Frequently asked questions
When does a matrix have no inverse?
When its determinant is zero. Such a matrix is singular — it collapses space onto a lower dimension, so the transformation can’t be reversed and no inverse exists.
What is the adjugate?
The adjugate is the transpose of the cofactor matrix — each entry replaced by its signed minor, then the whole matrix flipped across its diagonal. Dividing it by the determinant produces the inverse.
How do I check the inverse is right?
Multiply the original matrix by the inverse. If the result is the identity matrix (ones on the diagonal, zeros elsewhere), the inverse is correct.
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This calculator is for educational purposes. Double-check important results before acting on them.