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Math 2418 Linear Algebra Quiz \#2


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[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 $AB \not= BA$).  



$$\left[\matrix{
1 & 1   \cr
0 & 0  \cr }\right]
\left[\matrix{
0 & 1   \cr
0 & 0  \cr }\right] =
\left[\matrix{
0 & 1   \cr
0 & 0  \cr }\right]\not=
\left[\matrix{
0 & 0   \cr
0 & 0  \cr }\right] =
\left[\matrix{
0 & 1   \cr
0 & 0  \cr }\right]
\left[\matrix{
1 & 1   \cr
0 & 0  \cr }\right]$$

There are many other correct answers.


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

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\centerline{$\left[\matrix{ 
1 & 0 & 0 & 0 & 0 & -3 \cr
0 & 1 & 2 & 0 & 1 & 5  \cr
0 & 0 & 0 & 1 & -3 & 2  \cr }\right]$}


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

$$(-3, ~5 - 2s - t, ~s, ~2 + 3t, ~t)$$


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