1.
\(\left(\begin{array}{rrr}3\amp -7\amp 2 \end{array} \right)\left(\begin{array}{r}1\\3\\-4 \end{array} \right)\) and \(\left(\begin{array}{r}3\\-7\\2 \end{array} \right)\left(\begin{array}{rrr}1\amp 3\amp -4 \end{array} \right)\)
Section 2.9 Matrices: Homework
Exercises Exercises
Exercise Group.
Compute the matrix products.
2.
\(\left(\begin{array}{rrr}2\amp -3\amp 5 \end{array} \right)\left(\begin{array}{r}6\\4\\-2 \end{array} \right)\) and \(\left(\begin{array}{r}2\\-3\\5 \end{array} \right)\left(\begin{array}{rrr}6\amp 4\amp -2 \end{array} \right).\)
Exercise Group.
Multiply or explain why you cannot do so.
3.
\(\left(\begin{array}{rr}3\amp -1\\4\amp 2 \end{array} \right)\left(\begin{array}{rr}5\amp -2\\0\amp -1\\-3\amp 0 \end{array} \right)\)
4.
\(\left(\begin{array}{rr}1\amp -2\\3\amp -1 \end{array} \right)\left(\begin{array}{rr}7\amp 3\\0\amp -6\\-1\amp 4 \end{array} \right)\)
5.
\(\left(\begin{array}{rrr}5\amp -2\amp 7 \end{array} \right)\left(\begin{array}{rrr}0\amp -2\amp -1\\2\amp 0\amp -7\\12\amp -7\amp 3 \end{array} \right)\)
6.
\(\left(\begin{array}{rrr}2\amp -5\amp 4 \end{array} \right)\left(\begin{array}{rrr}1\amp 0\amp -5\\3\amp 1\amp -1\\11\amp 6\amp -2 \end{array} \right)\)
7.
\(\left(\begin{array}{rrr}5\amp -9\amp 7\\0\amp 6\amp -2\\4\amp 3\amp -1 \end{array} \right)\left(\begin{array}{r}2\\5\\-3 \end{array} \right)\)
8.
\(\left(\begin{array}{rrr}3\amp -2\amp -7\\1\amp 6\amp 1\\-4\amp 7\amp 5 \end{array} \right)\left(\begin{array}{r}2\\5\\-3 \end{array} \right)\)
9.
\(\left(\begin{array}{rrr}2\amp -5\amp 4 \end{array} \right)\left(\begin{array}{rrr}1\amp 0\amp -5\\3\amp 1\amp -1\\11\amp 6\amp -2 \end{array} \right)\)
Exercise Group.
Write each system of linear equations as a matrix-vector equation \(\A\vec{u}=\vec{v}.\)
10.
\begin{align*}
12a-b \amp = -5 \\
-2a+6b \amp = 10
\end{align*}
11.
\begin{align*}
2x-3y \amp +4z \amp =0\\
-x+6y\amp -2z \amp =4 \\
5x+y\amp \amp=9
\end{align*}
Exercise Group.
Write each matrix-vector equation as a system of linear equations.
12.
\(\lmatrix{rr} 8 \amp 2 \\ -1 \amp 3 \rmatrix \lmatrix{r} z \\w \rmatrix = \lmatrix{r} 2\\-2\rmatrix\)
13.
\(\lmatrix{rrrr} 2 \amp 0 \amp -1 \amp 6 \\ 3 \amp -1 \amp 0 \amp 2 \\ 0 \amp 0 \amp -1 \amp 14 \rmatrix \lmatrix{r} a\\b\\c\\d\rmatrix = \lmatrix{r} 53\\17\\51\rmatrix\)
14.
Find the matrix \(\A:=\begin{pmatrix}a_{11}\amp a_{12}\\a_{21}\amp a_{22} \end{pmatrix}\) for which \(\A\begin{pmatrix}x\\y \end{pmatrix} =\left(\begin{array}{r}5x-y\\2x+9y \end{array} \right)
.\)
15.
Find the matrix \(\A:=\begin{pmatrix}a_{11}\amp a_{12}\\a_{21}\amp a_{22} \end{pmatrix}\) for which \(\A\begin{pmatrix}x\\y \end{pmatrix} =\left(\begin{array}{r}-2x+4y\\3x-7y \end{array} \right)
.\)
16.
Find the matrix \(\B:=\begin{pmatrix}b_{11}\amp b_{12}\\b_{21}\amp b_{22} \end{pmatrix}\) for which \(\\B\begin{pmatrix}-3x+5y\\x-2y \end{pmatrix} =\begin{pmatrix}x\\y \end{pmatrix} .\)
17.
Let \(\C:=\begin{pmatrix}-2\amp 0\\\,\,0\amp 1/3 \end{pmatrix}\) and \(n\in\N=\{1,2,3,\ldots\}.\) Determine \(\C^n.\) (Do this by computing it for a few small values of \(n,\) determining the pattern, and making an argument about what must be the case when \(n\) is odd and what must be the case when \(n\) is even.) What is the geometric effect of the multiplication \(\C\begin{pmatrix}x\\y \end{pmatrix}\) on \(\begin{pmatrix}x\\y \end{pmatrix}\text{?}\)
18.
Repeat Problem 2.9.17 with \(\C=\left(\begin{array}{r}0\amp 1\\-1\amp 0 \end{array}\right).\)
19.
Let \(\boldsymbol{F}:=\begin{pmatrix}1\amp 1\\1\amp 0 \end{pmatrix} .\) Write the sequence
\begin{equation}
((F^1)_{11},(F^2)_{11},(F^3)_{11},(F^4)_{11},\ldots)\text{.}\tag{2.9.1}
\end{equation}
What do you notice?
20.
Find two different \(2\times 2\) matrices \(\A=\begin{pmatrix}a_{11}\amp a_{12}\\a_{21}\amp a_{22} \end{pmatrix}\) and \(\B=\begin{pmatrix}b_{11}\amp b_{12}\\b_{21}\amp b_{22} \end{pmatrix}\) which map \(\left(\begin{array}{r}4\\2\end{array}\right)\) to \(\begin{pmatrix}-1\\-2 \end{pmatrix} .\)
21.
Find two different \(2\times 2\) matrices \(\A=\begin{pmatrix}a_1\amp b_1\\c_1\amp d_1 \end{pmatrix}\) and \(\B=\begin{pmatrix}a_2\amp b_2\\c_2\amp d_2 \end{pmatrix}\) which map \(\left(\begin{array}{r}-3\\1\end{array}\right)\) to \(\begin{pmatrix}6\\3 \end{pmatrix} .\)
22.
for every \(x,y\in\R,\) left-multiplication by \(\A\) reflects \(\begin{pmatrix}x\\y \end{pmatrix}\) across the line \(\{(x,y)\in\R^2\,|\,y=-x\}=\{(x,-x)\in\R^2\}\)
23.
Find a \(2\times 2\) matrix \(\begin{pmatrix}a\amp b\\c\amp d \end{pmatrix}\) which rotates \(\begin{pmatrix}x\\y \end{pmatrix}\) by \(\displaystyle{\frac{11\pi}{6}}\) radians.
24.
Find a \(2\times 2\) matrix \(\begin{pmatrix}a\amp b\\c\amp d \end{pmatrix}\) which rotates \(\begin{pmatrix}x\\y \end{pmatrix}\) by \(\displaystyle{\frac{5\pi}{3}}\) radians.
25.
Prove Theorem 2.7.15.
