Exercise 8

When we explored matrix multiplication in Section 2.2, we saw that some properties that are true for real numbers are not true for matrices. This exercise will investigate that in some more depth.

  1. Suppose that \(A\) and \(B\) are two matrices and that \(AB = 0\text{.}\) If \(B \neq 0\text{,}\) what can you say about the linear independence of the columns of \(A\text{?}\)

  2. Suppose that we have matrices \(A\text{,}\) \(B\) and \(C\) such that \(AB = AC\text{.}\) We have seen that we cannot generally conclude that \(B=C\text{.}\) If we assume additionally that \(A\) is a matrix whose columns are linearly independent, explain why \(B = C\text{.}\) You may wish to begin by rewriting the equation \(AB = AC\) as \(AB-AC = A(B-C) = 0\text{.}\)