1.  Matrix Arithmetic

2.  Matrix Arithmetic & Linear Equations

3.  Matrix Arithmetic & Linear Maps

4.  Exercises 1 & 3

The linear system $$MX=Y$$ with $$M=\left(\begin{array}{ccc}\hfill 1& \hfill -1& \hfill 1\\ \hfill 5& \hfill -4& \hfill 3\\ \hfill 3& \hfill -3& \hfill 2\end{array}\right)X=\left(\begin{array}{c}\hfill x\\ \hfill y\\ \hfill z\end{array}\right)Y=\left(\begin{array}{c}\hfill u\\ \hfill v\\ \hfill w\end{array}\right)$$ has solution $$\left(\begin{array}{c}\hfill x\\ \hfill y\\ \hfill z\end{array}\right)=\left(\begin{array}{c}\hfill -u+v-w\\ \hfill u+v-2w\\ \hfill 3u-w\end{array}\right)$$ or $$\left(\begin{array}{c}\hfill x\\ \hfill y\\ \hfill z\end{array}\right)=\left(\begin{array}{ccc}\hfill -1& \hfill 1& \hfill -1\\ \hfill 1& \hfill 1& \hfill -2\\ \hfill 3& \hfill 0& \hfill -1\end{array}\right)\left(\begin{array}{c}\hfill u\\ \hfill v\\ \hfill w\end{array}\right)=Q\left(\begin{array}{c}\hfill u\\ \hfill v\\ \hfill w\end{array}\right)Q=\left(\begin{array}{ccc}\hfill -1& \hfill 1& \hfill -1\\ \hfill 1& \hfill 1& \hfill -2\\ \hfill 3& \hfill 0& \hfill -1\end{array}\right)$$

5.  Exercises 1 & 3 Retrospective

• The execution of row operations shows that $$X=QY\text{if}MX=Y$$

• Each elementary row operation can be reversed by another.

• $$MX=Y\text{if and only if}X=QY$$

• Morever, $$\left(MQ\right)Y=M\left(QY\right)=MX=Y\text{for every}Y=\left(\begin{array}{c}\hfill u\\ \hfill v\\ \hfill w\end{array}\right)$$

• So $$f_{MQ}=f_M\circ f_Q=\text{the identity map}\text{and}MQ=1=\left(\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1\end{array}\right)$$

6.  About invertible matrices

7.  Exercise No. 4

A slightly different matrix $$N=\left(\begin{array}{ccc}\hfill 1& \hfill -2& \hfill 1\\ \hfill 5& \hfill -4& \hfill 3\\ \hfill 3& \hfill -3& \hfill 2\end{array}\right)$$

• A square matrix

• Its row echelon forms have two non-zero rows and one zero row

• Row reduction for the linear system $$NX=Y=\left(\begin{array}{c}\hfill u\\ \hfill v\\ \hfill w\end{array}\right)\text{(arbitrary right-hand side, as above)}$$ leads to (in the last row) $$0=u+v-2w$$

• Only one of the three given right-hand sides admits solutions

• $$NX=\left(\begin{array}{c}\hfill 1\\ \hfill 1\\ \hfill 1\end{array}\right)\text{if and only if}X=\left(\begin{array}{c}\hfill -1\\ \hfill 0\\ \hfill 2\end{array}\right)+t\left(\begin{array}{c}\hfill -1\\ \hfill 1\\ \hfill 3\end{array}\right)\text{any}t$$