Overview
The inverse of a matrix A is a matrix A⁻¹ satisfying A × A⁻¹ = I, where I is the identity matrix. A 2×2 matrix has an inverse only when its determinant is nonzero; it is found by swapping the diagonal, changing the signs of the other entries, and dividing by the determinant.
Formula
A⁻¹ = (1/(ad − bc)) × [[d, −b], [−c, a]].
How it works
Compute the determinant
|A| = ad − bc.
Check invertibility
If the determinant is zero, no inverse exists.
Rearrange the entries
Swap a and d, and negate b and c.
Divide by the determinant
Multiply the resulting matrix by 1/|A|.
Worked examples
Example 1: A = [[4, 7], [2, 6]]
|A| = 10, and the inverse is [[0.6, −0.7], [−0.2, 0.4]].
Example 2: A = [[1, 2], [2, 4]]
|A| = 0, so no inverse exists.
Frequently asked questions
How do I verify the inverse?
Multiply A by the inverse; the result should be the identity matrix.
Does every matrix have an inverse?
No, only square matrices with a nonzero determinant.
What is a singular matrix?
A matrix with determinant zero, which is not invertible.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.