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2×2 Matrix Determinant & Inverse

Compute the determinant and inverse of a 2×2 matrix [[a, b], [c, d]], with a clear warning when the matrix is singular.

a (row 1, col 1)

b (row 1, col 2)

c (row 2, col 1)

d (row 2, col 2)

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How it works

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

For a 2×2 matrix [[a, b], [c, d]] the determinant is ad − bc, a single number that measures how the matrix scales area. When the determinant is non-zero the inverse exists and equals one over the determinant times the adjugate [[d, −b], [−c, a]] — swap the diagonal, negate the off-diagonal, divide by det. A zero determinant means the matrix is singular (its rows are linearly dependent) and cannot be inverted.

Worked example

For [[4, 7], [2, 6]], the determinant is 4·6 − 7·2 = 24 − 14 = 10. The inverse is (1/10)·[[6, −7], [−2, 4]] = [[0.6, −0.7], [−0.2, 0.4]]. Multiplying the two matrices returns the identity, confirming the result.

Sources: Invertible matrix — Wikipedia

Frequently asked questions

When does a 2×2 matrix have no inverse?

Exactly when its determinant ad − bc equals zero. Geometrically the matrix collapses the plane onto a line (or a point), so the transformation cannot be undone. Such a matrix is called singular, and the calculator flags it instead of dividing by zero.

What is the adjugate?

For a 2×2 matrix the adjugate is [[d, −b], [−c, a]] — you swap the two diagonal entries and flip the sign of the two off-diagonal entries. Dividing the adjugate by the determinant gives the inverse directly, which is why the formula is so compact for 2×2.

How can I check the inverse is correct?

Multiply the original matrix by its inverse. If the result is the identity matrix [[1, 0], [0, 1]], the inverse is right. Small rounding in the last decimal places is normal for non-integer entries.

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This calculator is for educational purposes. Double-check important results before acting on them.