The invertible matrix theorem

The invertible matrix theorem is a great result in linear algebra because based on the invertibility of a matrix, we can say a great many things about that matrix that are also true. Since we now know a quick way to determine the invertibility of a matrix through the computation of the determinant, we get all these other properties for free!

Here is the actual definition. Let A be a square n × n matrix. If A is invertible (det(A) ≠ 0), then the following properties follow:

• The column vectors of A are linearly independent.
• The column vectors of A form a basis for ℂⁿ.
• The row vectors of A form a basis for ℂⁿ and are also linearly independent.
• The linear transformation mapping |x⟩ to A|x⟩ is a bijection from ℂⁿ to ℂⁿ (we studied bijections in Anchor 3, Foundations).

If the matrix A is not invertible (det(A) = 0), then all the preceding properties are false.

While it may...