# Invertible matrix

From Conservapedia

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*. -
*A*x = b has a unique solution, for all b. -
*A*x = 0 has only the trivial solution of x = 0. -
*A*is row equivalent to an upper triangular matrix having non-zero diagonal entries.