- Theorem: the invertible matrix theorem.
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.
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
This is the content of this theorem in Section 3.5.
See this proposition in Section 3.5.
To reiterate, the invertible matrix theorem means:
There are two kinds of square matrices:
For invertible matrices, all of the statements of the invertible matrix theorem are true.
For non-invertible matrices, all of the statements of the invertible matrix theorem are false.
The reader should be comfortable translating any of the statements in the invertible matrix theorem into a statement about the pivots of a matrix.
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.
Now we can show that to check it's enough to show or
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
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.