Inverse of a 2×2 Matrix

Tap a term to see what it means, or .

Calculate

Example: solve the system 2x₁ + x₂ = 5 and x₁ + 3x₂ = 10. The coefficient matrix is A = (2 1; 1 3), and its inverse delivers the solution in one go. Leave y blank to compute just the inverse, or change the values to work out your own case.

a (row 1, column 1) b (row 1, column 2) c (row 2, column 1) d (row 2, column 2) y₁ (optional) y₂ (optional)

Step by step

What each term means

Inverse Matrix (A⁻¹): The matrix that undoes A: multiplied by A, it gives the identity. With it, the system A · x = y is solved in one step, x = A⁻¹ · y.
One Over the Determinant: The whole inverse is divided by the determinant. That is why it only exists when the determinant is not zero.
Determinant (ad − bc): The main diagonal minus the other one. It measures how much the matrix stretches areas: the unit square becomes a parallelogram of area |ad − bc|. If it is zero, the matrix flattens the plane onto a line and there is no way back.
Adjugate Matrix: The original matrix with the entries on the main diagonal swapped and the other two with their sign flipped. Multiplied by A, it gives the determinant times the identity.