Section 2.8 Matrices: In-Class Practice
Worksheet 2.8.1 Worksheet
2.
Prove or disprove: Matrix multiplication is commutative when exactly one of the matrices is diagonal.Exercise Group.
Let
\begin{equation*}
\A = \left(\begin{array}{rr}1\amp 2\\3\amp -1 \end{array} \right).
\end{equation*}
3.
Find a \(2 \times 2\) matrix \(\B\ne\boldsymbol{0}_{2 \times 2}\) so that \(\A\B \ne\B\A\) or explain why such a \(\B\) does not exist.
4.
Find a \(2 \times 2\) matrix \(\B\ne\boldsymbol{0}_{2 \times 2}\) so that \(\A\B=\B\A\) or explain why such a \(\B\) does not exist.
5.
Write the following system of linear equations as a matrix-vector equation \(\A\x=\b.\)
\begin{align*}
2x_1-3x_2-x_3+x_4 \amp =0\\
-x_1+2x_2-3x_4 \amp =10\\
2x_2-x_3+3x_4 \amp =1
\end{align*}
6.
Write the following matrix-vector equation as a system of linear equations:
\begin{equation*}
\lmatrix{rr} 2 \amp -3\\ 1 \amp -2 \rmatrix \cdot \lmatrix{r}x\\y \rmatrix = \lmatrix{r} -1\\1\rmatrix
\end{equation*}
7.
Multiply or explain why you can not do so: \(\left(\begin{array}{rrr}4\amp -1\amp -9\\4\amp 0\amp -2 \end{array} \right)\left(\begin{array}{rr}5\amp -3\\4\amp 0\\-3\amp 1 \end{array} \right)\)
8.
Multiply or explain why you can not do so: \(\left(\begin{array}{rrr}4\amp -1\amp -9\\3\amp 2\amp -1 \end{array} \right)\left(\begin{array}{rr}5\amp -3 \end{array} \right)\)
9.
Multiply or explain why you can not do so: \(\left(\begin{array}{rrr}2\amp -1\amp -7\\3\amp 5\amp -1\\6\amp 0\amp -2 \end{array} \right)\left(\begin{array}{r}3\\9\\-4 \end{array} \right)\)
Exercise Group.
For each part find a \(2\times 2\) matrix \(\A=\begin{pmatrix}a\amp b\\c\amp d \end{pmatrix}\) which maps the given vector as indicated.
10.
\(\A\) maps \(\begin{pmatrix}2\\3 \end{pmatrix}\) to \(\begin{pmatrix}1\\4 \end{pmatrix}\) (In other words \(\A \begin{pmatrix}2\\3 \end{pmatrix} =\begin{pmatrix}1\\4 \end{pmatrix}\)).
11.
for every \(x,y\in\R,\) \(\A\) maps \(\begin{pmatrix}x\\y \end{pmatrix}\) to \(\begin{pmatrix}x\\0 \end{pmatrix}\)
12.
for every \(x,y\in\R,\) \(\A\) maps \(\begin{pmatrix}x\\y \end{pmatrix}\) to \(\begin{pmatrix}0\\y \end{pmatrix}\)
13.
for every \(x,y\in\R,\) left-multiplication of \(\begin{pmatrix}x\\y\end{pmatrix}\) by \(\A\) acts as scalar multiplication by 2; i.e., maps \(\begin{pmatrix}x\\y \end{pmatrix}\) to \(\begin{pmatrix}2x\\2y \end{pmatrix}\)
14.
for every \(x,y\in\R,\) left-multiplication by \(\A\) reflects \(\begin{pmatrix}x\\y \end{pmatrix}\) across the line \(\{(x,y)\in\R^2\,|\,y=x\}=\{(x,x)\in\R^2\}\)
15.
for every \(x,y\in\R,\) left-multiplication by \(\A\) rotates \(\begin{pmatrix}x\\y\end{pmatrix}\) counterclockwise about the origin by \(\pi\) radians (using geometric considerations only, not the formula in Example 2.7.19)
16.
for every \(x,y\in\R,\) left-multiplication by \(\A\) rotates \(\begin{pmatrix}x\\y\end{pmatrix}\) counterclockwise about the origin by \(\displaystyle{\frac{3\pi}{2}}\) radians (using geometric considerations only, not the formula in Example 2.7.19)
Exercise Group.
(from Introduction to Linear Algebra by Gilbert Strang) Let \(\A\) and \(\B\) be arbitrary matrices. True or False, with justification (proof). If true explain why, if false give a counterexample.
17.
If \(\A^2\) is defined then \(\A\) is square.
18.
If \(\A\B\) and \(\B\A\) are defined then \(\A\) and \(\B\) are square.
19.
If \(\A\B\) and \(\B\A\) are defined then \(\A\B\) and \(\B\A\) are square.
20.
\((\A\B)^2=\A^2\B^2.\)
21.
If \(\A\B=\B\) then \(\A=I.\)
