#show link: set text(blue)
#set text(font: "Calibri")
#show raw: set text(font: "Fira Code")
#set table.cell(breakable: false)
#set table(stroke: (x, y) => (
  left: if x > 0 {
    .1pt
  },
  top: if y == 1 {
    0.5pt
  } else if y > 1 {
    0.1pt
  },
))


#set math.mat(delim: "[")
#set page(margin: (x: .5in, y: .5in))
#let solve(solution) = {
  block(
    inset: 10pt,
    stroke: blue + .3pt,
    fill: rgb(0, 149, 255, 15%),
    radius: 4pt,
  )[#solution]
}

#let solvein(solution) = {
  let outset = 3pt
  h(outset)
  box(
    outset: outset,
    stroke: blue + .3pt,
    fill: rgb(0, 149, 255, 15%),
    radius: 4pt,
  )[#solution]
}

#let note(content) = {
  block(
    outset: 5pt,
    inset: 10pt,
    stroke: luma(20%) + .3pt,
    fill: luma(95%),
    radius: 4pt,
  )[#content]
}

#let notein(content) = {
  let outset = 3pt
  h(outset)
  box(
    outset: outset,
    stroke: luma(20%) + .3pt,
    fill: luma(95%),
    radius: 4pt,
  )[#content]
}

#align(center)[
  = CS 3333 Mathematical Foundations
  Homework 4 (100 points)\
  #underline[Price Hiller] | #underline[zfp106]
]
#line(length: 100%, stroke: .25pt)

*Submission:*\
Same as HW1.

*Questions*\
Please write down the major intermediate steps.

1. Calculate the sum of two matrices if it is defined. (10 pts)

  #enum(
    numbering: "(a)",
    tight: false,
    number-align: start + top,
    [
      (5 pts)
      $mat(
        5, 7, -2;
        6, 0, 5;
        0, 4, 1;
      ) + mat(
        3, 0, 4;
        -5, -6, 8;
        7, 9, 0
      ) =$

      #note[
        $mat(
          5, 7, -2;
          6, 0, 5;
          0, 4, 1;
        ) + mat(
          3, 0, 4;
          -5, -6, 8;
          7, 9, 0
        ) &= mat(5 + 3, 7 + 0, 4 + (-2);6 + (-5), 0 + (-6), 5 + 8; 0 + 7, 4 + 9, 1 + 0;)\

        &= #solve[$mat(8, 7, 2; 1, -6, 13; 7, 13, 1;)$]$
      ]
    ],
    [
      (5 pts)
      $mat(
        2, 0, 7;
        9, 5, 6;
        8, 4, 0;
      ) + mat(
        6, 1;
        4, 7;
        0, 5;
      ) =$
      #solve[The addition is *not defined*. The first matrix is of order $3×3$ whereas the order of the second matrix is of order $3×2$.]
    ],
  )

2. Calculate $A*B$ if it is defined. (25 pts)\
  #enum(
    numbering: "(a)",
    tight: false,
    number-align: start + top,
    [
      (5 pts) $A = 8$, $B = mat(
        7, 0, -2, 4;
        -6, 1, -3, 5;
      )$

      #note[$
          "AB" &= 8 mat(
            7, 0, -2, 4;
            -6, 1, -3, 5;
          )\
          "AB" &= mat(
            8 ⋅ 7, 8 ⋅ 0, 8 ⋅ -2, 8 ⋅ 4;
            8 ⋅ -6, 8 ⋅ 1, 8 ⋅ -3, 8 ⋅ 5;
          )\
          "AB" &= #solve[$mat(
            56&, 0&, -16&, 32&;
            -48&, 8&, -24&, 40&;
          )$]
        $]
    ],
    [
      (10 pts) $A = mat(7, 0, -2, 4; -6, 1, -3, 5;)$ $B = mat(-1, 6; -4, 4; 5, 8; 0, -7;)$
      #note[
        $
          "AB" &= mat(7, 0, -2, 4; -6, 1, -3, 5;) × mat(-1, 6; -4, 4; 5, 8; 0, -7;)\
          "AB" &= mat(
            ((7 * -1) + (0 * -4) + (-2 * 5) + (4 * 0)), ((7 * 6) + (0 * 4) + (-2 * 8) + (4 * -7));
            ((-6 * -1) + (1 * -4) + (-3 * 5) + (5 * 0)), ((-6 * 6) + (1 * 4) + (-3 * 8) + (5 * -7));
          )\
          "AB" &= #solve[$mat(
            -17&, -2&;
            -13&, -91&;
          )$]
        $
      ]
    ],
    [
      (10 pts) $A = mat(1, 0, 0; 0, -1, -1; -1, 1, 0;)$ $B = mat(1, 1, -1; 0, -1, 1; 1, 1, 0;)$

      #note[
        $
          "AB" &= mat(1, 0, 0; 0, -1, -1; -1, 1, 0;) × mat(1, 1, -1; 0, -1, 1; 1, 1, 0;)\
          "AB" &= mat(
            ((1 * 1) + (0 * 0) + (0 * 1)), ((1 * 1) + (0 * -1) + (0 * 1)), ((1 * -1) + (0 * 1) + (0 * 0));
            ((0 * 1) + (-1 * 0) + (-1 * 1)), ((0 * 1) + (-1 * -1) + (-1 * 1)), ((0 * -1) + (-1 * 1) + (-1 * 0));
            ((-1 * 1) + (1 * 0) + (0 * 1)), ((-1 * 1) + (1 * -1) + (0 * 1)), ((-1 * -1) + (1 * 1) + (0 * 0))
          )\
          "AB" &= #solve[$mat(
            1&, 1&, -1&;
            -1, 0&, -1&;
            -1, -2&, 2&
          )$]\
        $
      ]
    ],
  )

3. Compute $"AB"$ and $"BA"$. Does $"AB" = "BA"$? (10 pts)

  $A = mat(2, 2; 2, 1;)$ and $B = mat(1, 2; 1, 2;)$

  #note[
    $
      "AB" &= mat(2, 2; 2, 1;) × mat(1, 2; 1, 2;)\
      "AB" &= mat(((2 * 1) + (2 * 1)), ((2 * 2) + (2 * 2)); ((2 * 1) + (1 * 1)), ((2 * 2) + (1 * 2));)\
      "AB" &= #solve[$mat(4, 8; 3, 6;)$]\
    $
    #align(center)[#line(length: 6cm)]
    $
      "BA" &= mat(1, 2; 1, 2;) × mat(2, 2; 2, 1;) \
      "BA" &= mat(((1 * 2) + (2 * 2)), ((1 * 2) + (2 * 1)); ((1 * 2) + (2 * 2)), ((1 * 2) + (2 * 1));)\
      "BA" &= #solve[$mat(6, 4; 6, 4;)$]\
    $

    #solve[$"AB" ≠ "BA"$]
  ]

4. Compute the transpose of matrix A (5 pts)

  $A = mat(9, 2, 5; 1, 0, 4;)$

  #solve[
    $
      A^t &= mat(9, 1; 2, 0; 5, 4)
    $
  ]

  #align(center)[#text(size: 2em)[#note[See next page]]]

5. #block(breakable: false)[Represent the following system of linear equations using matrices

    $
      a_11x_1 + a_12x_2 + a_13x_3 + a_14x_4 &= b_1\
      a_21x_1 + a_22x_2 + a_23x_3 + a_24x_4 &= b_2\
      a_31x_1 + a_32x_2 + a_33x_3 + a_34x_4 &= b_3\
      a_41x_1 + a_42x_2 + a_43x_3 + a_44x_4 &= b_4
    $

    The representation is $A*X = B$. What is matrices $A$, $X$, and $B$? (10 pts)

    #align(center)[
      #solve[
        #table(
          stroke: (x, y) => (
            left: none,
            top: if y > 0 {
              .5pt
            },
          ),
          columns: (auto, auto, auto, auto, auto),
          align: center + horizon,
          table.header([$A$], [], [$X$], [], [$B$]),
          [
            $
              mat(
            delim: "(",
            a_11, a_12, a_13, a_14;
            a_21, a_22, a_23, a_24;
            a_31, a_32, a_33, a_34;
            a_41, a_42, a_43, a_44;
          )
            $
          ],
          [$⋅$],
          [
            $
              mat(delim: "(",
            x_1;
            x_2;
            x_3;
            x_4;
          )
            $
          ],
          [$=$],
          [
            $
              mat(delim: "(",
            b_1;
            b_2;
            b_3;
            b_4;
          )
            $
          ],
        )
      ]
    ]
  ]

6. Show the adjacency matrix for the following graph. (20 pts)

  #figure(
    image("./assets/graph.png", width: 60%),
  ) <fig-graph>

  #align(center)[
    #solve[
      #table(
        stroke: (x, y) => (
          left: if y > 0 {
            if x == 1 {
              0.5pt
            } else if x > 1 {
              .1pt
            }
          },
          top: if x > 0 {
            if y == 1 {
              0.5pt
            } else if y > 1 {
              .1pt
            }
          },
        ),
        columns: (auto, auto, auto, auto, auto, auto, auto),
        fill: (x, y) => {
          if calc.odd(y) and y > 0 and x > 0 {
            color.hsl(200deg, 60%, 40%, 25%)
          } else {
            none
          }
        },
        inset: 3pt,
        [ ], [A], [B], [C], [D], [E], [F],
        [A], [1], [1], [0], [0], [1], [0],
        [B], [1], [1], [1], [1], [1], [0],
        [C], [0], [1], [1], [0], [1], [1],
        [D], [0], [1], [0], [1], [1], [0],
        [E], [1], [1], [1], [1], [1], [0],
        [F], [0], [0], [1], [0], [0], [1],
      )
      #notein[
        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.

        #text(size: .9em)[_Arguably though, that table _is_ easier to read._]
      ]
    ]
  ]

#align(center)[#text(size: 2em)[#note[See next page]]]

7. #block(breakable: false)[Compute the determinant of matrix $A$. (20 pts)
    $
      A = mat(
        6, 1, 4, 8;
        4, 2, 3, 2;
        4, 1, 2, 3;
        9, 7, 5, 6;
      )
    $
    #align(center)[
      #note[
        #align(left)[#note[
            $|A| = 6mat(
            2, 3, 2;
            1, 2, 3;
            7, 5, 6;
          ) - 1mat(
            4, 3, 2;
            4, 2, 3;
            9, 5, 6;
          ) + 4mat(
            4, 2, 2;
            4, 1, 3;
            9, 7, 6;
          ) - 8mat(
            4, 2, 3;
            4, 1, 2;
            9, 7, 5;
          )$
          ]]
        #align(left)[#note[
            #notein[The expanded values were found via *Sarrus' rule* for each $3 × 3$ matrix above.]
            $
              |A| &=\
              6&[
                (2 * 2 * 6) + (3 * 3 * 7) + (2 * 1 * 5) - (7 * 2 * 2) - (
                  5 * 3 * 2
                ) - (6 * 1 * 3)
              ]\
              - 1&[
                (4 * 2 * 6) + (3 * 3 * 9) + (2 * 4 * 5) - (9 * 2 * 2) - (
                  5 * 3 * 4
                ) - (6 * 4 * 3)
              ]\
              + 4&[
                (4 * 1 * 6) + (2 * 3 * 9) + (2 * 4 * 7) - (9 * 1 * 2) - (
                  7 * 3 * 4
                ) - (6 * 4 * 2)
              ]\
              - 8&[
                (4 * 1 * 5) + (2 * 2 * 9) + (3 * 4 * 7) - (9 * 1 * 3) - (
                  7 * 2 * 4
                ) - (5 * 4 * 2)
              ]\
            $
          ]]
        #align(left)[#note[
            $
              |A| &=
              6[21]
              - 1[1]
              + 4[-16]
              - 8[17]\
            $
          ]
        ]
        #align(center)[#solve[-75]]
      ]
    ]
  ]