Exercises 3 Perform the following matrix multiplications: $$ \begin{array}{c} {\left[\begin{array}{cccc} 8 & 15 & 19 & 4 \ 7 & -4 & 12 & 3 \end{array}\right] *\left[\begin{array}{cc} -15 & 19 \ -12 & -19 \ 0 & -13 \ 10 & 7 \end{array}\right]=\left[\begin{array}{ll} \square- \end{array}\right]=\left[\begin{array}{ll} - & - \ - & - \end{array}\right]} \ {\left[\begin{array}{cccc} 16 & 10 & -12 & -11 \ 10 & -15 & 15 & 5 \ -1 & 14 & -9 & 0 \ 3 & -3 & -16 & -5 \end{array}\right] *\left[\begin{array}{c} 4 \ 11 \ 6 \ 3 \end{array}\right]=\left[\begin{array}{ll} \square- \ - \ - \end{array}\right]=\left[\begin{array}{l} - \ - \ - \ - \end{array}\right]} \end{array} $$ Exercises 4 Perform the following multiplication of a matrix by a scalar $$ 6 *\left[\begin{array}{cc} 15 & 19 \ 2 & 5 \ 0 & -1 \ 1 & 7 \end{array}\right]=\left[\begin{array}{ll} - & - \ - & - \ - & - \end{array}\right] $$ Calculate the transpose of the following matrix $$ \left[\begin{array}{cc} 3 & 11 \ 6 & 4 \ 23 & 7 \end{array}\right]^{\top}=\left[\begin{array}{lll} - & - & - \ - & - & - \end{array}\right] $$

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**Exercise 3.1**

We wish to compute
$$
\begin{pmatrix}
8 & 15 & 19 & 4 \
7 & -4 & 12 & 3
\end{pmatrix}
\cdot
\begin{pmatrix}
-15 & 19 \
-12 & -19 \
0 & -13 \
10 & 7
\end{pmatrix}.
$$

*Step 1: Determine the size.*  
The first matrix is $$2 \times 4$$, and the second is $$4 \times 2$$, so the product will be a $$2 \times 2$$ matrix.

*Step 2: Compute each entry.*

- For the $$(1,1)$$ entry:
  $$
  8\cdot(-15) + 15\cdot(-12) + 19\cdot0 + 4\cdot10 = -120 - 180 + 0 + 40 = -260.
  $$

- For the $$(1,2)$$ entry:
  $$
  8\cdot19 + 15\cdot(-19) + 19\cdot(-13) + 4\cdot7 = 152 - 285 - 247 + 28 = -352.
  $$

- For the $$(2,1)$$ entry:
  $$
  7\cdot(-15) + (-4)\cdot(-12) + 12\cdot0 + 3\cdot10 = -105 + 48 + 0 + 30 = -27.
  $$

- For the $$(2,2)$$ entry:
  $$
  7\cdot19 + (-4)\cdot(-19) + 12\cdot(-13) + 3\cdot7 = 133 + 76 - 156 + 21 = 74.
  $$

*Step 3: Write the final answer.*
$$
\begin{pmatrix}
-260 & -352 \
-27 & 74
\end{pmatrix}.
$$

---

**Exercise 3.2**

We wish to compute
$$
\begin{pmatrix}
16 & 10 & -12 & -11 \
10 & -15 & 15 & 5 \
-1 & 14 & -9 & 0 \
3 & -3 & -16 & -5
\end{pmatrix}
\cdot
\begin{pmatrix}
4 \
11 \
6 \
3
\end{pmatrix}.
$$

*Step 1: Determine the size.*  
The first matrix is $$4 \times 4$$ and the column vector is $$4 \times 1$$, so the product is a $$4 \times 1$$ vector.

*Step 2: Compute each entry.*

- For the first entry:
  $$
  16\cdot4 + 10\cdot11 + (-12)\cdot6 + (-11)\cdot3 = 64 + 110 - 72 - 33 = 69.
  $$

- For the second entry:
  $$
  10\cdot4 + (-15)\cdot11 + 15\cdot6 + 5\cdot3 = 40 - 165 + 90 + 15 = -20.
  $$

- For the third entry:
  $$
  (-1)\cdot4 + 14\cdot11 + (-9)\cdot6 + 0\cdot3 = -4 + 154 - 54 + 0 = 96.
  $$

- For the fourth entry:
  $$
  3\cdot4 + (-3)\cdot11 + (-16)\cdot6 + (-5)\cdot3 = 12 - 33 - 96 - 15 = -132.
  $$

*Step 3: Write the final answer.*
$$
\begin{pmatrix}
69 \
-20 \
96 \
-132
\end{pmatrix}.
$$

---

**Exercise 4.1**

We wish to multiply the matrix by the scalar $$6$$:
$$
6 \cdot
\begin{pmatrix}
15 & 19 \
2 & 5 \
0 & -1 \
1 & 7
\end{pmatrix}.
$$

Multiply each entry by $$6$$:

- First row: $$15 \times 6 = 90$$ and $$19 \times 6 = 114$$.
- Second row: $$2 \times 6 = 12$$ and $$5 \times 6 = 30$$.
- Third row: $$0 \times 6 = 0$$ and $$-1 \times 6 = -6$$.
- Fourth row: $$1 \times 6 = 6$$ and $$7 \times 6 = 42$$.

Thus, the resulting matrix is
$$
\begin{pmatrix}
90 & 114 \
12 & 30 \
0 & -6 \
6 & 42
\end{pmatrix}.
$$

---

**Exercise 4.2**

We wish to compute the transpose of
$$
\begin{pmatrix}
3 & 11 \
6 & 4 \
23 & 7
\end{pmatrix}.
$$

*Step 1: Determine the size.*  
The given matrix is $$3 \times 2$$, so its transpose will be $$2 \times 3$$.

*Step 2: Write the transpose.*  
The first row becomes the first column and the second row becomes the second column:
$$
\begin{pmatrix}
3 & 6 & 23 \
11 & 4 & 7
\end{pmatrix}.
$$

---

**Final Answers**

- **Exercise 3.1:**
  $$
  \begin{pmatrix}
  -260 & -352 \
  -27 & 74
  \end{pmatrix}
  $$
- **Exercise 3.2:**
  $$
  \begin{pmatrix}
  69 \
  -20 \
  96 \
  -132
  \end{pmatrix}
  $$
- **Exercise 4.1:**
  $$
  6 \cdot
  \begin{pmatrix}
  15 & 19 \
  2 & 5 \
  0 & -1 \
  1 & 7
  \end{pmatrix}
  =
  \begin{pmatrix}
  90 & 114 \
  12 & 30 \
  0 & -6 \
  6 & 42
  \end{pmatrix}
  $$
- **Exercise 4.2:**
  $$
  \begin{pmatrix}
  3 & 11 \
  6 & 4 \
  23 & 7
  \end{pmatrix}^{\top}
  =
  \begin{pmatrix}
  3 & 6 & 23 \
  11 & 4 & 7
  \end{pmatrix}
  $$
**Exercise 3.1:** $$ \begin{pmatrix} -260 & -352 \ -27 & 74 \end{pmatrix} $$ **Exercise 3.2:** $$ \begin{pmatrix} 69 \ -20 \ 96 \ -132 \end{pmatrix} $$ **Exercise 4.1:** $$ \begin{pmatrix} 90 & 114 \ 12 & 30 \ 0 & -6 \ 6 & 42 \end{pmatrix} $$ **Exercise 4.2:** $$ \begin{pmatrix} 3 & 6 & 23 \ 11 & 4 & 7 \end{pmatrix} $$

Exercises 3

Perform the following matrix multiplications:

Exercises 4
Perform the following multiplication of a matrix by a scalar

Calculate the transpose of the following matrix

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Exercises 3 Perform the following matrix multiplications: Exercises 4 Perform the following multiplication of a matrix by a scalar Calculate the transpose of the following matrix