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+#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]]
+      ]
+    ]
+  ]
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