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2x2 Matrix Inverse Calculator

Enter a 2×2 matrix to compute its inverse.

How it is calculated

The matrix is entered row by row: the top row is a, b and the bottom row is c, d.

det = a·d − b·c
inverse = (1 ÷ det) × [[d, −b], [−c, a]]

In other words, swap a and d, change the signs of b and c, then divide all four numbers by the determinant. The tool shows the determinant and the two rows of the inverse, each as a pair separated by a comma and a space. Numbers are shown with up to 4 decimals, with trailing zeros removed, so 1/3 appears as 0.3333. If the determinant is 0, or so close to 0 that it is only floating-point noise next to the products a·d and b·c, the matrix is treated as singular and there is no result. All four entries are required.

Worked example

Matrix [[4, 7], [2, 6]]. Determinant: 4 × 6 − 7 × 2 = 24 − 14 = 10. Swap the diagonal and negate the others to get [[6, −7], [−2, 4]], then divide by 10. First row: 6 ÷ 10 = 0.6 and −7 ÷ 10 = −0.7. Second row: −2 ÷ 10 = −0.2 and 4 ÷ 10 = 0.4. Check: 4 × 0.6 + 7 × (−0.2) = 2.4 − 1.4 = 1, as the identity matrix requires.

Try this example →

Common mistakes

  • Entering the columns instead of the rows. The first two boxes (a, b) are the top row; swapping b and c gives the inverse of the transposed matrix.
  • Forgetting that the off-diagonal entries change sign. In the answer, b and c appear negated and a and d swap places.
  • Expecting an answer when the rows are multiples of each other, such as [[2, 4], [1, 2]]. Its determinant is 0, so there is no inverse and nothing is shown.
  • Reading the answer as one list of four numbers. It is two rows: the first pair is the top row and the second pair is the bottom row.
  • Treating a rounded entry as exact. An entry like 0.3333 is 1/3 cut to four decimals, so multiplying back gives 0.9999 rather than exactly 1.

About this calculator

The inverse of a 2×2 matrix [[a, b], [c, d]] is given by 1/det × [[d, −b], [−c, a]], where the determinant det = ad − bc. Multiplying a matrix by its inverse always produces the identity matrix, and the inverse only exists when the determinant is non-zero — a zero determinant means the matrix is singular and collapses space in a way that cannot be undone.

Matrix inverses are how systems of two linear equations in two unknowns get solved directly, converting 'ax + by = e, cx + dy = f' into a single matrix equation that the inverse solves in one step, which is the matrix-algebra equivalent of Cramer's rule. The same formula underlies the classical Hill cipher in cryptography, where a message is encrypted by multiplying letter values by a matrix and decrypted by multiplying by its inverse, and it is a routine step in computer graphics whenever a transformation — rotation, scaling, shearing — needs to be undone.

Enter the four entries of a 2×2 matrix, and the calculator computes its inverse when one exists.

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