forked from TeamBasedInquiryLearning/linear-algebra
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathM1.html
415 lines (415 loc) · 25.3 KB
/
M1.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
<!DOCTYPE html>
<!--********************************************-->
<!--* Generated from PreTeXt source *-->
<!--* on 2021-01-14T10:40:38-06:00 *-->
<!--* A recent stable commit (2020-08-09): *-->
<!--* 98f21740783f166a773df4dc83cab5293ab63a4a *-->
<!--* *-->
<!--* https://pretextbook.org *-->
<!--* *-->
<!--********************************************-->
<html lang="en-US">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Matrices and Multiplication (M1)</title>
<meta name="Keywords" content="Authored in PreTeXt">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
tex2jax: {
inlineMath: [['\\(','\\)']]
},
asciimath2jax: {
ignoreClass: ".*",
processClass: "has_am"
},
jax: ["input/AsciiMath"],
extensions: ["asciimath2jax.js"],
TeX: {
extensions: ["extpfeil.js", "autobold.js", "https://pretextbook.org/js/lib/mathjaxknowl.js", "AMScd.js", ],
// scrolling to fragment identifiers is controlled by other Javascript
positionToHash: false,
equationNumbers: { autoNumber: "none", useLabelIds: true, },
TagSide: "right",
TagIndent: ".8em",
},
// HTML-CSS output Jax to be dropped for MathJax 3.0
"HTML-CSS": {
scale: 88,
mtextFontInherit: true,
},
CommonHTML: {
scale: 88,
mtextFontInherit: true,
},
});
</script><script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-AMS_CHTML-full"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/lib/jquery.min.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/lib/jquery.sticky.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/lib/jquery.espy.min.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/0.13/pretext.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/0.13/pretext_add_on.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script xmlns:svg="http://www.w3.org/2000/svg">sagecellEvalName='Evaluate (Sage)';
</script><link xmlns:svg="http://www.w3.org/2000/svg" href="https://fonts.googleapis.com/css?family=Open+Sans:400,400italic,600,600italic" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://fonts.googleapis.com/css?family=Inconsolata:400,700&subset=latin,latin-ext" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/pretext.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/pretext_add_on.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/banner_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/toc_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/knowls_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/style_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/colors_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/setcolors.css" rel="stylesheet" type="text/css">
<!-- 2019-10-12: Temporary - CSS file for experiments with styling --><link xmlns:svg="http://www.w3.org/2000/svg" href="developer.css" rel="stylesheet" type="text/css">
</head>
<body class="mathbook-book has-toc has-sidebar-left">
<a class="assistive" href="#content">Skip to main content</a><div xmlns:svg="http://www.w3.org/2000/svg" class="hidden-content" style="display:none">\(\require{enclose}
\newcommand{\IR}{\mathbb{R}}
\newcommand{\IC}{\mathbb{C}}
\renewcommand{\P}{\mathcal{P}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\RREF}{\operatorname{RREF}}
\newcommand{\vspan}{\operatorname{span}}
\newcommand{\setList}[1]{\left\{#1\right\}}
\newcommand{\setBuilder}[2]{\left\{#1\,\middle|\,#2\right\}}
\newcommand{\unknown}{\,{\color{gray}?}\,}
\newcommand{\circledNumber}[1]{\enclose{circle}{#1}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\)</div>
<header id="masthead" class="smallbuttons"><div class="banner"><div class="container">
<a id="logo-link" href=""></a><div class="title-container">
<h1 class="heading"><a href="linear-algebra-for-team-based-inquiry-learning.html"><span class="title">Linear Algebra for Team-Based Inquiry Learning</span></a></h1>
<p class="byline">Steven Clontz, Drew Lewis</p>
</div>
</div></div>
<nav xmlns:svg="http://www.w3.org/2000/svg" id="primary-navbar" class="navbar"><div class="container">
<div class="navbar-top-buttons">
<button class="sidebar-left-toggle-button button active" aria-label="Show or hide table of contents sidebar">Contents</button><div class="tree-nav toolbar toolbar-divisor-3"><span class="threebuttons"><a id="previousbutton" class="previous-button toolbar-item button" href="M.html" title="Previous">Prev</a><a id="upbutton" class="up-button button toolbar-item" href="M.html" title="Up">Up</a><a id="nextbutton" class="next-button button toolbar-item" href="M2.html" title="Next">Next</a></span></div>
</div>
<div class="navbar-bottom-buttons toolbar toolbar-divisor-4">
<button class="sidebar-left-toggle-button button toolbar-item active">Contents</button><a class="previous-button toolbar-item button" href="M.html" title="Previous">Prev</a><a class="up-button button toolbar-item" href="M.html" title="Up">Up</a><a class="next-button button toolbar-item" href="M2.html" title="Next">Next</a>
</div>
</div></nav></header><div class="page">
<div xmlns:svg="http://www.w3.org/2000/svg" id="sidebar-left" class="sidebar" role="navigation"><div class="sidebar-content">
<nav id="toc"><ul>
<li class="link frontmatter"><a href="frontmatter.html" data-scroll="frontmatter"><span class="title">Front Matter</span></a></li>
<li class="link">
<a href="E.html" data-scroll="E"><span class="codenumber">1</span> <span class="title">Systems of Linear Equations (E)</span></a><ul>
<li><a href="E1.html" data-scroll="E1">Linear Systems, Vector Equations, and Augmented Matrices (E1)</a></li>
<li><a href="E2.html" data-scroll="E2">Row Reduction of Matrices (E2)</a></li>
<li><a href="E3.html" data-scroll="E3">Solving Linear Systems (E3)</a></li>
</ul>
</li>
<li class="link">
<a href="V.html" data-scroll="V"><span class="codenumber">2</span> <span class="title">Vector Spaces (V)</span></a><ul>
<li><a href="V1.html" data-scroll="V1">Vector Spaces (V1)</a></li>
<li><a href="V2.html" data-scroll="V2">Linear Combinations (V2)</a></li>
<li><a href="V3.html" data-scroll="V3">Spanning Sets (V3)</a></li>
<li><a href="V4.html" data-scroll="V4">Subspaces (V4)</a></li>
<li><a href="V5.html" data-scroll="V5">Linear Independence (V5)</a></li>
<li><a href="V6.html" data-scroll="V6">Identifying a Basis (V6)</a></li>
<li><a href="V7.html" data-scroll="V7">Subspace Basis and Dimension (V7)</a></li>
<li><a href="V8.html" data-scroll="V8">Polynomial and Matrix Spaces (V8)</a></li>
<li><a href="V9.html" data-scroll="V9">Homogeneous Linear Systems (V9)</a></li>
</ul>
</li>
<li class="link">
<a href="A.html" data-scroll="A"><span class="codenumber">3</span> <span class="title">Algebraic Properties of Linear Maps (A)</span></a><ul>
<li><a href="A1.html" data-scroll="A1">Linear Transformations (A1)</a></li>
<li><a href="A2.html" data-scroll="A2">Standard Matrices (A2)</a></li>
<li><a href="A3.html" data-scroll="A3">Image and Kernel (A3)</a></li>
<li><a href="A4.html" data-scroll="A4">Injective and Surjective Linear Maps (A4)</a></li>
</ul>
</li>
<li class="link">
<a href="M.html" data-scroll="M"><span class="codenumber">4</span> <span class="title">Matrices (M)</span></a><ul>
<li><a href="M1.html" data-scroll="M1" class="active">Matrices and Multiplication (M1)</a></li>
<li><a href="M2.html" data-scroll="M2">Row Operations as Matrix Multiplication (M2)</a></li>
<li><a href="M3.html" data-scroll="M3">The Inverse of a Matrix (M3)</a></li>
<li><a href="M4.html" data-scroll="M4">Invertible Matrices (M4)</a></li>
</ul>
</li>
<li class="link">
<a href="G.html" data-scroll="G"><span class="codenumber">5</span> <span class="title">Geometric Properties of Linear Maps (G)</span></a><ul>
<li><a href="G1.html" data-scroll="G1">Row Operations and Determinants (G1)</a></li>
<li><a href="G2.html" data-scroll="G2">Computing Determinants (G2)</a></li>
<li><a href="G3.html" data-scroll="G3">Eigenvalues and Characteristic Polynomials (G3)</a></li>
<li><a href="G4.html" data-scroll="G4">Eigenvectors and Eigenspaces (G4)</a></li>
</ul>
</li>
</ul></nav><div class="extras"><nav><a class="mathbook-link" href="https://pretextbook.org">Authored in PreTeXt</a><a href="https://www.mathjax.org"><img title="Powered by MathJax" src="https://www.mathjax.org/badge/badge.gif" alt="Powered by MathJax"></a></nav></div>
</div></div>
<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="M1"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">4.1</span> <span class="title">Matrices and Multiplication (M1)</span>
</h2>
<article class="observation remark-like" id="observation-22"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">4.1.1</span><span class="period">.</span>
</h6>
<p id="p-686">If \(T: \IR^n \rightarrow \IR^m\) and \(S: \IR^m \rightarrow \IR^k\) are linear maps, then the composition map \(S\circ T\) is a linear map from \(\IR^n \rightarrow \IR^k\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzcd}[ampersand replacement=\&]
\IR^n \arrow[rr, bend right, "S\circ T"'] \arrow[r,"T"] \& \IR^m \arrow[r,"S"] \&\IR^k
\end{tikzcd}
</pre>
<p id="p-687">Recall that for a vector, \(\vec{v} \in \IR^n\text{,}\) the composition is computed as \((S \circ T)(\vec{v})=S(T(\vec{v}))\text{.}\)</p></article><article class="activity project-like" id="activity-105"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">4.1.1</span><span class="period">.</span>
</h6>
<p id="p-688">Let \(T: \IR^3 \rightarrow \IR^2\) be given by the \(2\times 3\) standard matrix \(B=\left[\begin{array}{c} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right]\) and \(S: \IR^2 \rightarrow \IR^4\) be given by the \(4\times 2\) standard matrix \(A=\left[\begin{array}{c} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}\)</p>
<p id="p-689">What are the domain and codomain of the composition map \(S \circ T\text{?}\)</p>
<ol class="lower-alpha">
<li id="li-509"><p id="p-690">The domain is \(\IR ^2\) and the codomain is \(\IR^3\)</p></li>
<li id="li-510"><p id="p-691">The domain is \(\IR ^3\) and the codomain is \(\IR^2\)</p></li>
<li id="li-511"><p id="p-692">The domain is \(\IR ^2\) and the codomain is \(\IR^4\)</p></li>
<li id="li-512"><p id="p-693">The domain is \(\IR ^3\) and the codomain is \(\IR^4\)</p></li>
<li id="li-513"><p id="p-694">The domain is \(\IR ^4\) and the codomain is \(\IR^3\)</p></li>
<li id="li-514"><p id="p-695">The domain is \(\IR ^4\) and the codomain is \(\IR^2\)</p></li>
</ol></article><article class="activity project-like" id="activity-106"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">4.1.2</span><span class="period">.</span>
</h6>
<p id="p-696">Let \(T: \IR^3 \rightarrow \IR^2\) be given by the \(2\times 3\) standard matrix \(B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right]\) and \(S: \IR^2 \rightarrow \IR^4\) be given by the \(4\times 2\) standard matrix \(A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}\)</p>
<p id="p-697">What size will the standard matrix of \(S \circ T:\IR^3\to\IR^4\) be? (Rows \(\times\) Columns)</p>
<ol class="lower-alpha">
<li id="li-515">\(\displaystyle 4 \times 3\)</li>
<li id="li-516">\(\displaystyle 4 \times 2\)</li>
<li id="li-517">\(\displaystyle 3 \times 4\)</li>
<li id="li-518">\(\displaystyle 3 \times 2\)</li>
<li id="li-519">\(\displaystyle 2 \times 4\)</li>
<li id="li-520">\(\displaystyle 2 \times 3\)</li>
</ol></article><article class="activity project-like" id="activity-107"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">4.1.3</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-30"><p id="p-698">Let \(T: \IR^3 \rightarrow \IR^2\) be given by the \(2\times 3\) standard matrix \(B=\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right]\) and \(S: \IR^2 \rightarrow \IR^4\) be given by the \(4\times 2\) standard matrix \(A=\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]\text{.}\)</p></div>
<article class="task exercise-like" id="task-70"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-699">Compute</p>
<div class="displaymath">
\begin{equation*}
(S \circ T)(\vec{e}_1)
=
S(T(\vec{e}_1))
=
S\left(\left[\begin{array}{c} 2 \\ 5\end{array}\right]\right)
=
\left[\begin{array}{c}\unknown\\\unknown\\\unknown\\\unknown\end{array}\right].
\end{equation*}
</div></article><article class="task exercise-like" id="task-71"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-700">Compute \((S \circ T)(\vec{e}_2)
\text{.}\)</p></article><article class="task exercise-like" id="task-72"><h6 class="heading"><span class="codenumber">(c)</span></h6>
<p id="p-701">Compute \((S \circ T)(\vec{e}_3)
\text{.}\)</p></article><article class="task exercise-like" id="task-73"><h6 class="heading"><span class="codenumber">(d)</span></h6>
<p id="p-702">Write the \(4\times 3\) standard matrix of \(S \circ T:\IR^3\to\IR^4\text{.}\)</p></article></article><article class="definition definition-like" id="definition-24"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">4.1.2</span><span class="period">.</span>
</h6>
<p id="p-703">We define the <dfn class="terminology">product</dfn> \(AB\) of a \(m \times n\) matrix \(A\) and a \(n \times k\) matrix \(B\) to be the \(m \times k\) standard matrix of the composition map of the two corresponding linear functions.</p>
<p id="p-704">For the previous activity, \(T\) was a map \(\IR^3 \rightarrow \IR^2\text{,}\) and \(S\) was a map \(\IR^2 \rightarrow \IR^4\text{,}\) so \(S \circ T\) gave a map \(\IR^3 \rightarrow \IR^4\) with a \(4\times 3\) standard matrix:</p>
<div class="displaymath">
\begin{equation*}
AB
=
\left[\begin{array}{cc} 1 & 2 \\ 0 & 1 \\ 3 & 5 \\ -1 & -2 \end{array}\right]
\left[\begin{array}{ccc} 2 & 1 & -3 \\ 5 & -3 & 4 \end{array}\right]
\end{equation*}
</div>
<div class="displaymath">
\begin{equation*}
=
\left[
(S \circ T)(\vec{e}_1) \hspace{1em}
(S\circ T)(\vec{e}_2) \hspace{1em}
(S \circ T)(\vec{e}_3)
\right]
=
\left[\begin{array}{ccc}
12 & -5 & 5 \\
5 & -3 & 4 \\
31 & -12 & 11 \\
-12 & 5 & -5
\end{array}\right]
.
\end{equation*}
</div></article><article class="activity project-like" id="activity-108"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">4.1.4</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-31"><p id="p-705">Let \(S: \IR^3 \rightarrow \IR^2\) be given by the matrix \(A=\left[\begin{array}{ccc} -4 & -2 & 3 \\ 0 & 1 & 1 \end{array}\right]\) and \(T: \IR^2 \rightarrow \IR^3\) be given by the matrix \(B=\left[\begin{array}{cc} 2 & 3 \\ 1 & -1 \\ 0 & -1 \end{array}\right]\text{.}\)</p></div>
<article class="task exercise-like" id="task-74"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-706">Write the dimensions (rows \(\times\) columns) for \(A\text{,}\) \(B\text{,}\) \(AB\text{,}\) and \(BA\text{.}\)</p></article><article class="task exercise-like" id="task-75"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-707">Find the standard matrix \(AB\) of \(S \circ T\text{.}\)</p></article><article class="task exercise-like" id="task-76"><h6 class="heading"><span class="codenumber">(c)</span></h6>
<p id="p-708">Find the standard matrix \(BA\) of \(T \circ S\text{.}\)</p></article></article><article class="activity project-like" id="activity-109"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">4.1.5</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-32">
<p id="p-709">Consider the following three matrices.</p>
<div class="displaymath">
\begin{equation*}
A = \left[\begin{array}{ccc}1&0&-3\\3&2&1\end{array}\right]
\hspace{2em}
B = \left[\begin{array}{ccccc}2&2&1&0&1\\1&1&1&-1&0\\0&0&3&2&1\\-1&5&7&2&1\end{array}\right]
\hspace{2em}
C = \left[\begin{array}{cc}2&2\\0&-1\\3&1\\4&0\end{array}\right]
\end{equation*}
</div>
</div>
<article class="task exercise-like" id="task-77"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-710">Find the domain and codomain of each of the three linear maps corresponding to \(A\text{,}\) \(B)\text{,}\) and \(C\text{.}\)</p></article><article class="task exercise-like" id="task-78"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-711">Only one of the matrix products \(AB,AC,BA,BC,CA,CB\) can actually be computed. Compute it.</p></article></article><article class="activity project-like" id="activity-110"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">4.1.6</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-33"><p id="p-712">Let \(B=\left[\begin{array}{ccc} 3 & -4 & 0 \\ 2 & 0 & -1 \\ 0 & -3 & 3 \end{array}\right]\text{,}\) and let \(A=\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]\text{.}\)</p></div>
<article class="task exercise-like" id="task-79"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-713">Compute the product \(BA\) by hand.</p></article><article class="task exercise-like" id="task-80"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-714">Check your work using technology. Using Octave:</p>
<pre class="code-block tex2jax_ignore">
B = sym([3 -4 0 ; 2 0 -1 ; 0 -3 3])
A = sym([2 7 -1 ; 0 3 2 ; 1 1 -1])
B*A
</pre></article></article><section class="exercises" id="exercises-17"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">4.1.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-81"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-715">Of the following three matrices, only two may be multiplied.</p>
<div class="displaymath">
\begin{align*}
A= \left[\begin{array}{cc}
-1 & 4 \\
4 & 5 \\
-2 & 3
\end{array}\right] & & B= \left[\begin{array}{ccc}
1 & 6 & -2 \\
-1 & 4 & 5 \\
0 & 3 & 1 \\
1 & 1 & -2
\end{array}\right] & & C= \left[\begin{array}{cc}
3 & -2 \\
2 & -1 \\
-1 & 0 \\
0 & -4
\end{array}\right]
\end{align*}
</div>
<p data-braille="continuation">Explain which two can be multiplied and why. Then show how to find their product.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-81" id="answer-81"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-81"><div class="answer solution-like"><div class="displaymath">
\begin{equation*}
BA= \left[\begin{array}{cc}
27 & 28 \\
7 & 31 \\
10 & 18 \\
7 & 3
\end{array}\right]
\end{equation*}
</div></div></div>
</div></article><article class="exercise exercise-like" id="exercise-82"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-716">Of the following three matrices, only two may be multiplied.</p>
<div class="displaymath">
\begin{align*}
A= \left[\begin{array}{cccc}
1 & 4 & -1 & 0 \\
-1 & -4 & 2 & -1
\end{array}\right] & & B= \left[\begin{array}{ccc}
1 & -1 & -4 \\
2 & -1 & -5
\end{array}\right] & & C= \left[\begin{array}{cccc}
1 & -5 & 5 & 5 \\
1 & -4 & 3 & 3 \\
1 & -5 & 6 & 6
\end{array}\right]
\end{align*}
</div>
<p data-braille="continuation">Explain which two can be multiplied and why. Then show how to find their product.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-82" id="answer-82"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-82"><div class="answer solution-like"><div class="displaymath">
\begin{equation*}
BC= \left[\begin{array}{cccc}
-4 & 19 & -22 & -22 \\
-4 & 19 & -23 & -23
\end{array}\right]
\end{equation*}
</div></div></div>
</div></article><article class="exercise exercise-like" id="exercise-83"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-717">Of the following three matrices, only two may be multiplied.</p>
<div class="displaymath">
\begin{align*}
A= \left[\begin{array}{cccc}
1 & 1 & -3 & -2 \\
0 & 1 & -4 & -1
\end{array}\right] & & B= \left[\begin{array}{cccc}
1 & 1 & 0 & 1 \\
-2 & -1 & -1 & -1 \\
-1 & 3 & -3 & 5
\end{array}\right] & & C= \left[\begin{array}{cc}
1 & -3 \\
-1 & 4 \\
-1 & 4
\end{array}\right]
\end{align*}
</div>
<p data-braille="continuation">Explain which two can be multiplied and why. Then show how to find their product.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-83" id="answer-83"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-83"><div class="answer solution-like"><div class="displaymath">
\begin{equation*}
CA= \left[\begin{array}{cccc}
1 & -2 & 9 & 1 \\
-1 & 3 & -13 & -2 \\
-1 & 3 & -13 & -2
\end{array}\right]
\end{equation*}
</div></div></div>
</div></article><article class="exercise exercise-like" id="exercise-84"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-718">Of the following three matrices, only two may be multiplied.</p>
<div class="displaymath">
\begin{align*}
A= \left[\begin{array}{cccc}
1 & -3 & 4 & 6 \\
0 & 1 & -2 & -1
\end{array}\right] & & B= \left[\begin{array}{ccc}
1 & 5 & 6 \\
0 & 0 & 1
\end{array}\right] & & C= \left[\begin{array}{ccc}
1 & -1 & -5 \\
0 & 1 & 3 \\
1 & -1 & -4 \\
0 & 0 & 2
\end{array}\right]
\end{align*}
</div>
<p data-braille="continuation">Explain which two can be multiplied and why. Then show how to find their product.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-84" id="answer-84"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-84"><div class="answer solution-like"><div class="displaymath">
\begin{equation*}
AC= \left[\begin{array}{ccc}
5 & -8 & -18 \\
-2 & 3 & 9
\end{array}\right]
\end{equation*}
</div></div></div>
</div></article><article class="exercise exercise-like" id="exercise-85"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-719">Of the following three matrices, only two may be multiplied.</p>
<div class="displaymath">
\begin{align*}
A= \left[\begin{array}{ccc}
2 & -1 & -1 \\
1 & 0 & -2 \\
3 & -1 & -2 \\
2 & -1 & -1
\end{array}\right] & & B= \left[\begin{array}{cc}
3 & 3 \\
0 & 1 \\
-2 & -2 \\
1 & 0
\end{array}\right] & & C= \left[\begin{array}{ccc}
3 & 5 & -3 \\
1 & 2 & -1
\end{array}\right]
\end{align*}
</div>
<p data-braille="continuation">Explain which two can be multiplied and why. Then show how to find their product.</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-85" id="answer-85"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-85"><div class="answer solution-like"><div class="displaymath">
\begin{equation*}
BC= \left[\begin{array}{ccc}
12 & 21 & -12 \\
1 & 2 & -1 \\
-8 & -14 & 8 \\
3 & 5 & -3
\end{array}\right]
\end{equation*}
</div></div></div>
</div></article><p id="p-720"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/M1/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
</div>
</body>
</html>