398 lines
9.2 KiB
Plaintext
398 lines
9.2 KiB
Plaintext
#show link: set text(blue)
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#set text(font: "Calibri")
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#show raw: set text(font: "Fira Code")
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#set table.cell(breakable: false)
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#set table(stroke: (x, y) => (
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left: if x > 0 {
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.1pt
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},
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top: if y == 1 {
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0.5pt
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} else if y > 1 {
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0.1pt
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},
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))
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#set math.mat(delim: "[")
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#set page(margin: (x: .5in, y: .5in))
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#let solve(solution) = {
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block(
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inset: 10pt,
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stroke: blue + .3pt,
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fill: rgb(0, 149, 255, 15%),
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radius: 4pt,
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)[#solution]
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}
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#let solvein(solution) = {
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let outset = 3pt
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h(outset)
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box(
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outset: outset,
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stroke: blue + .3pt,
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fill: rgb(0, 149, 255, 15%),
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radius: 4pt,
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)[#solution]
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}
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#let note(content) = {
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block(
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outset: 5pt,
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inset: 10pt,
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stroke: luma(20%) + .3pt,
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fill: luma(95%),
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radius: 4pt,
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)[#content]
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}
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#let notein(content) = {
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let outset = 3pt
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h(outset)
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box(
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outset: outset,
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stroke: luma(20%) + .3pt,
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fill: luma(95%),
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radius: 4pt,
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)[#content]
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}
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#align(center)[
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= CS 3333 Mathematical Foundations
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Homework 4 (100 points)\
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#underline[Price Hiller] | #underline[zfp106]
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]
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#line(length: 100%, stroke: .25pt)
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*Submission:*\
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Same as HW1.
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*Questions*\
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Please write down the major intermediate steps.
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1. Calculate the sum of two matrices if it is defined. (10 pts)
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#enum(
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numbering: "(a)",
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tight: false,
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number-align: start + top,
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[
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(5 pts)
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$mat(
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5, 7, -2;
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6, 0, 5;
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0, 4, 1;
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) + mat(
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3, 0, 4;
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-5, -6, 8;
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7, 9, 0
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) =$
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#note[
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$mat(
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5, 7, -2;
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6, 0, 5;
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0, 4, 1;
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) + mat(
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3, 0, 4;
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-5, -6, 8;
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7, 9, 0
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) &= mat(5 + 3, 7 + 0, 4 + (-2);6 + (-5), 0 + (-6), 5 + 8; 0 + 7, 4 + 9, 1 + 0;)\
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&= #solve[$mat(8, 7, 2; 1, -6, 13; 7, 13, 1;)$]$
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]
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],
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[
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(5 pts)
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$mat(
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2, 0, 7;
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9, 5, 6;
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8, 4, 0;
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) + mat(
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6, 1;
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4, 7;
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0, 5;
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) =$
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#solve[The addition is *not defined*. The first matrix is of order $3×3$ whereas the order of the second matrix is of order $3×2$.]
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],
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)
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2. Calculate $A*B$ if it is defined. (25 pts)\
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#enum(
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numbering: "(a)",
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tight: false,
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number-align: start + top,
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[
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(5 pts) $A = 8$, $B = mat(
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7, 0, -2, 4;
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-6, 1, -3, 5;
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)$
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#note[$
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"AB" &= 8 mat(
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7, 0, -2, 4;
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-6, 1, -3, 5;
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)\
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"AB" &= mat(
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8 ⋅ 7, 8 ⋅ 0, 8 ⋅ -2, 8 ⋅ 4;
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8 ⋅ -6, 8 ⋅ 1, 8 ⋅ -3, 8 ⋅ 5;
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)\
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"AB" &= #solve[$mat(
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56&, 0&, -16&, 32&;
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-48&, 8&, -24&, 40&;
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)$]
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$]
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],
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[
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(10 pts) $A = mat(7, 0, -2, 4; -6, 1, -3, 5;)$ $B = mat(-1, 6; -4, 4; 5, 8; 0, -7;)$
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#note[
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$
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"AB" &= mat(7, 0, -2, 4; -6, 1, -3, 5;) × mat(-1, 6; -4, 4; 5, 8; 0, -7;)\
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"AB" &= mat(
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((7 * -1) + (0 * -4) + (-2 * 5) + (4 * 0)), ((7 * 6) + (0 * 4) + (-2 * 8) + (4 * -7));
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((-6 * -1) + (1 * -4) + (-3 * 5) + (5 * 0)), ((-6 * 6) + (1 * 4) + (-3 * 8) + (5 * -7));
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)\
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"AB" &= #solve[$mat(
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-17&, -2&;
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-13&, -91&;
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)$]
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$
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]
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],
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[
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(10 pts) $A = mat(1, 0, 0; 0, -1, -1; -1, 1, 0;)$ $B = mat(1, 1, -1; 0, -1, 1; 1, 1, 0;)$
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#note[
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$
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"AB" &= mat(1, 0, 0; 0, -1, -1; -1, 1, 0;) × mat(1, 1, -1; 0, -1, 1; 1, 1, 0;)\
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"AB" &= mat(
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((1 * 1) + (0 * 0) + (0 * 1)), ((1 * 1) + (0 * -1) + (0 * 1)), ((1 * -1) + (0 * 1) + (0 * 0));
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((0 * 1) + (-1 * 0) + (-1 * 1)), ((0 * 1) + (-1 * -1) + (-1 * 1)), ((0 * -1) + (-1 * 1) + (-1 * 0));
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((-1 * 1) + (1 * 0) + (0 * 1)), ((-1 * 1) + (1 * -1) + (0 * 1)), ((-1 * -1) + (1 * 1) + (0 * 0))
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)\
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"AB" &= #solve[$mat(
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1&, 1&, -1&;
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-1, 0&, -1&;
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-1, -2&, 2&
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)$]\
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$
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]
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],
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)
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3. Compute $"AB"$ and $"BA"$. Does $"AB" = "BA"$? (10 pts)
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$A = mat(2, 2; 2, 1;)$ and $B = mat(1, 2; 1, 2;)$
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#note[
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$
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"AB" &= mat(2, 2; 2, 1;) × mat(1, 2; 1, 2;)\
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"AB" &= mat(((2 * 1) + (2 * 1)), ((2 * 2) + (2 * 2)); ((2 * 1) + (1 * 1)), ((2 * 2) + (1 * 2));)\
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"AB" &= #solve[$mat(4, 8; 3, 6;)$]\
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$
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#align(center)[#line(length: 6cm)]
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$
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"BA" &= mat(1, 2; 1, 2;) × mat(2, 2; 2, 1;) \
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"BA" &= mat(((1 * 2) + (2 * 2)), ((1 * 2) + (2 * 1)); ((1 * 2) + (2 * 2)), ((1 * 2) + (2 * 1));)\
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"BA" &= #solve[$mat(6, 4; 6, 4;)$]\
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$
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#solve[$"AB" ≠ "BA"$]
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]
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4. Compute the transpose of matrix A (5 pts)
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$A = mat(9, 2, 5; 1, 0, 4;)$
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#solve[
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$
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A^t &= mat(9, 1; 2, 0; 5, 4)
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$
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]
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#align(center)[#text(size: 2em)[#note[See next page]]]
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5. #block(breakable: false)[Represent the following system of linear equations using matrices
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$
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a_11x_1 + a_12x_2 + a_13x_3 + a_14x_4 &= b_1\
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a_21x_1 + a_22x_2 + a_23x_3 + a_24x_4 &= b_2\
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a_31x_1 + a_32x_2 + a_33x_3 + a_34x_4 &= b_3\
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a_41x_1 + a_42x_2 + a_43x_3 + a_44x_4 &= b_4
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$
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The representation is $A*X = B$. What is matrices $A$, $X$, and $B$? (10 pts)
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#align(center)[
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#solve[
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#table(
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stroke: (x, y) => (
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left: none,
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top: if y > 0 {
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.5pt
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},
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),
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columns: (auto, auto, auto, auto, auto),
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align: center + horizon,
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table.header([$A$], [], [$X$], [], [$B$]),
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[
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$
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mat(
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delim: "(",
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a_11, a_12, a_13, a_14;
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a_21, a_22, a_23, a_24;
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a_31, a_32, a_33, a_34;
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a_41, a_42, a_43, a_44;
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)
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$
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],
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[$⋅$],
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[
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$
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mat(delim: "(",
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x_1;
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x_2;
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x_3;
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x_4;
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)
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$
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],
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[$=$],
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[
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$
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mat(delim: "(",
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b_1;
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b_2;
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b_3;
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b_4;
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)
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$
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],
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)
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]
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]
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]
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6. Show the adjacency matrix for the following graph. (20 pts)
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#figure(
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image("./assets/graph.png", width: 60%),
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) <fig-graph>
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#align(center)[
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#solve[
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#table(
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stroke: (x, y) => (
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left: if y > 0 {
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if x == 1 {
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0.5pt
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} else if x > 1 {
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.1pt
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}
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},
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top: if x > 0 {
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if y == 1 {
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0.5pt
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} else if y > 1 {
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.1pt
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}
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},
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),
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columns: (auto, auto, auto, auto, auto, auto, auto),
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fill: (x, y) => {
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if calc.odd(y) and y > 0 and x > 0 {
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color.hsl(200deg, 60%, 40%, 25%)
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} else {
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none
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}
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},
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inset: 3pt,
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[ ], [A], [B], [C], [D], [E], [F],
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[A], [1], [1], [0], [0], [1], [0],
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[B], [1], [1], [1], [1], [1], [0],
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[C], [0], [1], [1], [0], [1], [1],
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[D], [0], [1], [0], [1], [1], [0],
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[E], [1], [1], [1], [1], [1], [0],
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[F], [0], [0], [1], [0], [0], [1],
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)
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#notein[
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Sorry about the lack of _proper_ matrix notation. I had a hard time typesetting the row and column labels for matrices in particular in Typst :(. I have to wait on https://github.com/typst/typst/issues/445 to get resolved I guess.
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#text(size: .9em)[_Arguably though, that table _is_ easier to read._]
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]
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]
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]
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#align(center)[#text(size: 2em)[#note[See next page]]]
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7. #block(breakable: false)[Compute the determinant of matrix $A$. (20 pts)
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$
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A = mat(
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6, 1, 4, 8;
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4, 2, 3, 2;
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4, 1, 2, 3;
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9, 7, 5, 6;
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)
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$
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#align(center)[
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#note[
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#align(left)[#note[
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$|A| = 6mat(
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2, 3, 2;
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1, 2, 3;
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7, 5, 6;
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) - 1mat(
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4, 3, 2;
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4, 2, 3;
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9, 5, 6;
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) + 4mat(
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4, 2, 2;
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4, 1, 3;
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9, 7, 6;
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) - 8mat(
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4, 2, 3;
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4, 1, 2;
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9, 7, 5;
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)$
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]]
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#align(left)[#note[
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#notein[The expanded values were found via *Sarrus' rule* for each $3 × 3$ matrix above.]
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$
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|A| &=\
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6&[
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(2 * 2 * 6) + (3 * 3 * 7) + (2 * 1 * 5) - (7 * 2 * 2) - (
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5 * 3 * 2
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) - (6 * 1 * 3)
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]\
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- 1&[
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(4 * 2 * 6) + (3 * 3 * 9) + (2 * 4 * 5) - (9 * 2 * 2) - (
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5 * 3 * 4
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) - (6 * 4 * 3)
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]\
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+ 4&[
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(4 * 1 * 6) + (2 * 3 * 9) + (2 * 4 * 7) - (9 * 1 * 2) - (
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7 * 3 * 4
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) - (6 * 4 * 2)
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]\
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- 8&[
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(4 * 1 * 5) + (2 * 2 * 9) + (3 * 4 * 7) - (9 * 1 * 3) - (
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7 * 2 * 4
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) - (5 * 4 * 2)
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]\
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$
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]]
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#align(left)[#note[
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$
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|A| &=
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6[21]
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- 1[1]
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+ 4[-16]
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- 8[17]\
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$
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]
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]
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#align(center)[#solve[-75]]
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]
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]
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]
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