Math 2418 Linear Algebra Quiz #2

[10] 1.) Prove that matrix multiplication for SQUARE matrices is not commutative (I.e., give two specific square matrices with real numbers and show with these matrices AB ≠ BA).

[ \begin{matrix}1 & 1 \\ 0 & 0 \end{matrix} ] [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] = [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] ≠ [ \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ] = [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] [ \begin{matrix}1 & 1 \\ 0 & 0 \end{matrix} ]

There are many other correct answers.

[10] 2.) Given the following augmented matrix, solve

[ \begin{matrix}1 & 0 & 0 & 0 & 0 & 8 \\ 0 & 1 & 2 & 0 & 1 & −5 \\ 0 & 0 & 0 & 1 & −3 & 4 \end{matrix} ]

Since the above metrix is in REF, we know the answer is

(8, −5 − 2s − t, s, 4 + 3t, t)