This section consists of a single important theorem containing many equivalent conditions for a matrix to be invertible. This is one of the most important theorems in this textbook. We will append two more criteria in Section 5.1.

Invertible Matrix Theorem

Let be an matrix, and let be the matrix transformation The following statements are equivalent:

The matrix has pivots if and only if its reduced row echelon form is the identity matrix This happens exactly when the procedure in Section 3.5 to compute the inverse succeeds.

The null space of a matrix is if and only if the matrix has no free variables, which means that every column is a pivot column, which means has pivots. See this recipe in Section 2.6.

We know has at least one solution for every if and only if the columns of span by this theorem in Section 3.2, and has at most one solution for every if and only if the columns of are linearly independent by this theorem in Section 3.2. Hence has exactly one solution for every if and only if its columns are linearly independent and span

The following conditions are also equivalent to the invertibility of a square matrix They are all simple restatements of conditions in the invertible matrix theorem.

The reduced row echelon form of is the identity matrix

has no solutions other than the trivial one.

The columns of form a basis for

is consistent for all in

Now we can show that to check it's enough to show or

Corollary(A Left or Right Inverse Suffices)

Let be an matrix, and suppose that there exists an matrix such that or Then is invertible and

Suppose that We claim that is onto. Indeed, for any in we have

so and hence is in the range of Therefore, is invertible by the invertible matrix theorem. Since is invertible, we have

so

Now suppose that We claim that is one-to-one. Indeed, suppose that Then so But so and hence Therefore, is invertible by the invertible matrix theorem. One shows that as above.

We conclude with some common situations in which the invertible matrix theorem is useful.