How to Use the Matrix Inverse Calculator
The inverse of a matrix A is a matrix A⁻¹ such that multiplying them together gives the identity matrix (A × A⁻¹ = I). An inverse exists only when the determinant is non-zero. For a 2×2 matrix [[a,b],[c,d]], the inverse is 1/(ad-bc) × [[d,-b],[-c,a]]. A 3×3 matrix's inverse is found by dividing the transpose of its cofactor matrix (the adjugate) by the determinant.
For example, matrix [[4,7],[2,6]] has a determinant of 4×6-7×2 = 10, so its inverse exists: 1/10 × [[6,-7],[-2,4]] = [[0.6,-0.7],[-0.2,0.4]]. If the determinant is exactly 0, the matrix is called singular and has no inverse — meaning the linear transformation it represents can't be reversed.
Matrix inverses are essential for solving systems of equations written as Ax=b (solved as x = A⁻¹b), reversing coordinate transformations in 3D graphics, and computing regression coefficients in statistics. Pick a matrix size, enter each element, and this calculator shows the determinant and the inverse together.
Frequently Asked Questions
The inverse exists only when the determinant is non-zero. If the determinant is 0, no inverse exists.
For matrix [[a,b],[c,d]], the inverse is 1/(ad-bc) × [[d,-b],[-c,a]].
It's used to solve systems of equations in matrix form, reverse linear transformations, and compute coordinate transforms in computer graphics.