An invertible matrix A is one that has a corresponding matrix B such that:
- AB = BA = I (I: identity matrix)
A matrix is invertible if and only if it has a nonzero determinant. Only square matrices (n x n) are invertible.
It can be shown that, for any invertible matrix A, there is only one matrix B such that AB = I; this unique matrix is called the inverse matrix of A, and represented as A-1.
There are five conditions that are equivalent to a matrix A being invertible:
- The determinant of A is not zero.
- A is row equivalent to I.
- Ax = b has a unique solution, for all b.
- Ax = 0 has only the trivial solution of x = 0.
- A is row equivalent to an upper triangular matrix having non-zero diagonal entries.