Matrix multiplication practice problems on a worksheet.

Let's solve each matrix multiplication problem step by step. Matrix multiplication involves taking the dot product of rows from the first matrix with columns from the second matrix.

---

(1)

$$
\begin{bmatrix}
0 & 0 \\
-5 & 5
\end{bmatrix}
\cdot
\begin{bmatrix}
-1 & 0 \\
-4 & 3
\end{bmatrix}
$$

Step-by-step:

We multiply row 1 of the first matrix with column 1 and column 2 of the second matrix:

- Row 1 × Col 1: $ (0)(-1) + (0)(-4) = 0 $
- Row 1 × Col 2: $ (0)(0) + (0)(3) = 0 $

- Row 2 × Col 1: $ (-5)(-1) + (5)(-4) = 5 - 20 = -15 $
- Row 2 × Col 2: $ (-5)(0) + (5)(3) = 0 + 15 = 15 $

Result:
$$
\begin{bmatrix}
0 & 0 \\
-15 & 15
\end{bmatrix}
$$

---

(2)

$$
\begin{bmatrix}
6 & 3 \\
2 & -1
\end{bmatrix}
\cdot
\begin{bmatrix}
3 & 2 \\
-4 & 8
\end{bmatrix}
$$

- Row 1 × Col 1: $ 6(3) + 3(-4) = 18 - 12 = 6 $
- Row 1 × Col 2: $ 6(2) + 3(8) = 12 + 24 = 36 $
- Row 2 × Col 1: $ 2(3) + (-1)(-4) = 6 + 4 = 10 $
- Row 2 × Col 2: $ 2(2) + (-1)(8) = 4 - 8 = -4 $

Result:
$$
\begin{bmatrix}
6 & 36 \\
10 & -4
\end{bmatrix}
$$

---

(3)

$$
\begin{bmatrix}
-3 & 5 \\
-2 & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
6 & -2 \\
1 & -5
\end{bmatrix}
$$

- Row 1 × Col 1: $ (-3)(6) + (5)(1) = -18 + 5 = -13 $
- Row 1 × Col 2: $ (-3)(-2) + (5)(-5) = 6 - 25 = -19 $
- Row 2 × Col 1: $ (-2)(6) + (1)(1) = -12 + 1 = -11 $
- Row 2 × Col 2: $ (-2)(-2) + (1)(-5) = 4 - 5 = -1 $

Result:
$$
\begin{bmatrix}
-13 & -19 \\
-11 & -1
\end{bmatrix}
$$

---

(4)

$$
\begin{bmatrix}
-1 \\
2 \\
3
\end{bmatrix}
\cdot
\begin{bmatrix}
2 & 4 & 3
\end{bmatrix}
$$

This is a 3×1 matrix times a 1×3 matrix → result is 3×3.

Multiply each element of the column vector with each element of the row vector:

- First row: $ (-1) \times [2, 4, 3] = [-2, -4, -3] $
- Second row: $ 2 \times [2, 4, 3] = [4, 8, 6] $
- Third row: $ 3 \times [2, 4, 3] = [6, 12, 9] $

Result:
$$
\begin{bmatrix}
-2 & -4 & -3 \\
4 & 8 & 6 \\
6 & 12 & 9
\end{bmatrix}
$$

---

(5)

$$
\begin{bmatrix}
6 & 5 \\
0 & -2 \\
-5 & -6
\end{bmatrix}
\cdot
\begin{bmatrix}
2 & 1 \\
-1 & 2
\end{bmatrix}
$$

This is a 3×2 matrix times a 2×2 matrix → result is 3×2

Compute each entry:

- Row 1 × Col 1: $ 6(2) + 5(-1) = 12 - 5 = 7 $
- Row 1 × Col 2: $ 6(1) + 5(2) = 6 + 10 = 16 $
- Row 2 × Col 1: $ 0(2) + (-2)(-1) = 0 + 2 = 2 $
- Row 2 × Col 2: $ 0(1) + (-2)(2) = 0 - 4 = -4 $
- Row 3 × Col 1: $ (-5)(2) + (-6)(-1) = -10 + 6 = -4 $
- Row 3 × Col 2: $ (-5)(1) + (-6)(2) = -5 -12 = -17 $

Result:
$$
\begin{bmatrix}
7 & 16 \\
2 & -4 \\
-4 & -17
\end{bmatrix}
$$

---

(6)

$$
\begin{bmatrix}
-6 & -4 \\
-2 & -2 \\
-5 & 4 \\
-4 & -3
\end{bmatrix}
\cdot
\begin{bmatrix}
6 & -3 \\
6 & 0
\end{bmatrix}
$$

This is a 4×2 matrix times a 2×2 matrix → result is 4×2

Compute each entry:

- Row 1 × Col 1: $ (-6)(6) + (-4)(6) = -36 -24 = -60 $
- Row 1 × Col 2: $ (-6)(-3) + (-4)(0) = 18 + 0 = 18 $
- Row 2 × Col 1: $ (-2)(6) + (-2)(6) = -12 -12 = -24 $
- Row 2 × Col 2: $ (-2)(-3) + (-2)(0) = 6 + 0 = 6 $
- Row 3 × Col 1: $ (-5)(6) + (4)(6) = -30 + 24 = -6 $
- Row 3 × Col 2: $ (-5)(-3) + (4)(0) = 15 + 0 = 15 $
- Row 4 × Col 1: $ (-4)(6) + (-3)(6) = -24 -18 = -42 $
- Row 4 × Col 2: $ (-4)(-3) + (-3)(0) = 12 + 0 = 12 $

Result:
$$
\begin{bmatrix}
-60 & 18 \\
-24 & 6 \\
-6 & 15 \\
-42 & 12
\end{bmatrix}
$$

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✔ Final Answers:

1. $$
\begin{bmatrix}
0 & 0 \\
-15 & 15
\end{bmatrix}
$$

2. $$
\begin{bmatrix}
6 & 36 \\
10 & -4
\end{bmatrix}
$$

3. $$
\begin{bmatrix}
-13 & -19 \\
-11 & -1
\end{bmatrix}
$$

4. $$
\begin{bmatrix}
-2 & -4 & -3 \\
4 & 8 & 6 \\
6 & 12 & 9
\end{bmatrix}
$$

5. $$
\begin{bmatrix}
7 & 16 \\
2 & -4 \\
-4 & -17
\end{bmatrix}
$$

6. $$
\begin{bmatrix}
-60 & 18 \\
-24 & 6 \\
-6 & 15 \\
-42 & 12
\end{bmatrix}
$$

Let me know if you'd like these verified or explained further!