Find the inverse of a 2×2 or 3×3 matrix. Enter the matrix with rows separated by semicolons.
Usage Tip
Not every square matrix is invertible — only those with a non-zero determinant (the non-singular ones).
2×2: swap the diagonal, negate the off-diagonal, divide by the determinant
Only square matrices with a non-zero determinant have an inverse.
A matrix with determinant zero is singular and has no inverse.
Multiplying a matrix by its inverse yields the identity.
The inverse of a matrix is the matrix that gives the identity matrix when multiplied by the original. It exists only when the determinant is not zero. A matrix with a zero determinant is called singular, and no inverse exists for it.
For a 2 by 2 matrix, the inverse swaps the diagonal entries, changes the signs of the other two, and divides everything by the determinant. For 3 by 3 matrices the calculation uses cofactors and is much longer by hand, which is where a calculator saves time. Inverses are used for solving systems of linear equations and for undoing transformations. Multiply the result by the original to check that you get the identity matrix.