2×2 Matrix Inverse
Inverse of a 2×2 matrix with determinant check
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Working out inverse of a 2×2 matrix with determinant check is easier with 2×2 Matrix Inverse, which answers as you type. Fill in a, b and c on 2×2 Matrix Inverse and read your answer immediately. 2×2 Matrix Inverse displays each intermediate figure as well. Use 2×2 Matrix Inverse for a reliable number, fast. 2×2 Matrix Inverse is privacy-first: Math inputs stay offline. 2×2 Matrix Inverse sits in the Math section next to free online 2×2 matrix inverse calculator and matrix inverse calculator 2x2 determinant. The labels in 2×2 Matrix Inverse match the terms in its result. 2×2 Matrix Inverse is free, instant and local.
What the defaults give
The matrix [[4, 7], [2, 6]] has determinant 10, so its inverse is [[0.6, −0.7], [−0.2, 0.4]].
Multiplying the two together returns the identity [[1, 0], [0, 1]], which is the check that the answer is right.
The formula
For [[a, b], [c, d]]:
A⁻¹ = 1 ÷ (ad − bc) × [[d, −b], [−c, a]]
Two steps. First the determinant ad − bc; if it is zero the matrix has no inverse, because dividing by zero is impossible and geometrically the transformation collapses the plane onto a line. Then swap the diagonal entries, negate the off-diagonal ones, and divide everything by the determinant.
Worked examples
[[4, 7], [2, 6]] → det 10, [[0.6, −0.7], [−0.2, 0.4]]
[[2, 7], [2, 6]] → det −2, [[−3.0, 3.5], [1.0, −1.0]]
[[8, 7], [2, 6]] → det 34, [[0.18, −0.21], [−0.06, 0.24]]
[[4, 3.5], [2, 6]] → det 17, [[0.35, −0.21], [−0.12, 0.24]]
[[4, 14], [2, 6]] → det −4, [[−1.5, 3.5], [0.5, −1.0]]
[[4, 7], [1, 6]] → det 17, [[0.35, −0.41], [−0.06, 0.24]]
Det 10 and det 17 both appear twice in that table with different entries, which is a reminder that the determinant is a single number summarising the whole matrix — many matrices share one.
What the determinant means
Geometrically, the determinant is the factor by which the matrix scales area. A determinant of 10 means a unit square becomes ten square units; a determinant of −2 means it becomes two units and the orientation flips, which is what a negative sign always indicates. Near zero, the transformation squashes everything toward a line, and the inverse becomes numerically unstable even when it technically exists.
When to be careful
Det exactly zero means singular: no inverse. Det very small means an inverse that exists but amplifies rounding error enormously, so a computer may produce a matrix that fails the identity check in the third decimal place. That is a property of the problem, not of the arithmetic.
Integer matrices with determinant ±1 have integer inverses — useful in puzzles and in exact arithmetic. Everything else produces fractions, which this page prints to two decimals; multiply the result by the determinant to recover the exact form when you need it.
Frequently Asked Questions
What is the inverse of [[4, 7], [2, 6]]?
[[0.6, −0.7], [−0.2, 0.4]]. The determinant is 4×6 − 7×2 = 10, and each entry is divided by it.
When does a 2×2 matrix have no inverse?
When ad − bc = 0. The matrix is singular, the transformation collapses area, and no matrix can undo it.
How do I check the answer?
Multiply A by A⁻¹. The result must be the identity matrix with 1s on the diagonal and 0s elsewhere.
Why do the signs flip?
The formula swaps the diagonal entries and negates b and c. That rearrangement is what makes the product come out as the identity.
What does a negative determinant mean?
The transformation reverses orientation — a reflection is involved. The magnitude still gives the area scaling factor.
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