How this calculator works
An inverse matrix undoes the transformation represented by A, so A times A inverse is the identity matrix. A square matrix has an inverse exactly when its determinant is non-zero. This calculator forms the adjugate and divides every entry by the determinant. For larger or nearly singular matrices, numerical elimination methods are usually preferred because the values can be sensitive to rounding.
Formula
A inverse = adj(A) / det(A). For 2 by 2, the inverse of a, b; c, d is (1/(ad - bc)) times d, -b; -c, a.
Worked example
The matrix with rows 1, 2 and 3, 4 has determinant -2 and inverse -2, 1; 1.5, -0.5. Multiplying the two matrices gives the 2 by 2 identity.
Advertisement
Frequently asked questions
Which matrices have an inverse?
Only square matrices with a non-zero determinant. Rectangular matrices have no two-sided inverse, although they can have a pseudoinverse.
What if the determinant is very small?
The matrix is close to singular and its inverse has very large entries. Small input changes can then produce large output changes, so treat the result cautiously.
Does a matrix commute with its inverse?
Yes. A times A inverse and A inverse times A both equal the identity, so this is one of the cases where the order does not matter.