Algebra 2 worksheet on matrix multiplication with four problems to solve.

To solve the given matrix multiplication problems, we need to follow the rules of matrix multiplication. Specifically:

1. Matrix Multiplication Rule: If \( A \) is an \( m \times n \) matrix and \( B \) is an \( n \times p \) matrix, then the product \( AB \) is an \( m \times p \) matrix. The element in the \( i \)-th row and \( j \)-th column of the resulting matrix is obtained by taking the dot product of the \( i \)-th row of \( A \) and the \( j \)-th column of \( B \).

Let's solve each problem step by step.

---

Problem 1:

\[
\begin{bmatrix}
3 & -5 \\
-4 & 2
\end{bmatrix}
*
\begin{bmatrix}
-2 & 0 \\
5 & -6
\end{bmatrix}
\]

#### Step 1: Dimensions Check
- The first matrix is \( 2 \times 2 \).
- The second matrix is \( 2 \times 2 \).
- The result will be a \( 2 \times 2 \) matrix.

#### Step 2: Compute Each Element
The resulting matrix \( C \) is computed as follows:
\[
C_{ij} = \text{(row } i \text{ of first matrix)} \cdot \text{(column } j \text{ of second matrix)}
\]

- Element \( C_{11} \):
\[
C_{11} = (3)(-2) + (-5)(5) = -6 - 25 = -31
\]

- Element \( C_{12} \):
\[
C_{12} = (3)(0) + (-5)(-6) = 0 + 30 = 30
\]

- Element \( C_{21} \):
\[
C_{21} = (-4)(-2) + (2)(5) = 8 + 10 = 18
\]

- Element \( C_{22} \):
\[
C_{22} = (-4)(0) + (2)(-6) = 0 - 12 = -12
\]

#### Final Result:
\[
\begin{bmatrix}
-31 & 30 \\
18 & -12
\end{bmatrix}
\]

---

Problem 2:

\[
\begin{bmatrix}
-2 & 7 \\
4 & -5
\end{bmatrix}
*
\begin{bmatrix}
-7 & 9 \\
4 & -3
\end{bmatrix}
\]

#### Step 1: Dimensions Check
- The first matrix is \( 2 \times 2 \).
- The second matrix is \( 2 \times 2 \).
- The result will be a \( 2 \times 2 \) matrix.

#### Step 2: Compute Each Element

- Element \( C_{11} \):
\[
C_{11} = (-2)(-7) + (7)(4) = 14 + 28 = 42
\]

- Element \( C_{12} \):
\[
C_{12} = (-2)(9) + (7)(-3) = -18 - 21 = -39
\]

- Element \( C_{21} \):
\[
C_{21} = (4)(-7) + (-5)(4) = -28 - 20 = -48
\]

- Element \( C_{22} \):
\[
C_{22} = (4)(9) + (-5)(-3) = 36 + 15 = 51
\]

#### Final Result:
\[
\begin{bmatrix}
42 & -39 \\
-48 & 51
\end{bmatrix}
\]

---

Problem 3:

\[
\begin{bmatrix}
-9 & 2 \\
8 & -1 \\
4 & -3
\end{bmatrix}
*
\begin{bmatrix}
-2 & 5 \\
3 & -6
\end{bmatrix}
\]

#### Step 1: Dimensions Check
- The first matrix is \( 3 \times 2 \).
- The second matrix is \( 2 \times 2 \).
- The result will be a \( 3 \times 2 \) matrix.

#### Step 2: Compute Each Element

- Element \( C_{11} \):
\[
C_{11} = (-9)(-2) + (2)(3) = 18 + 6 = 24
\]

- Element \( C_{12} \):
\[
C_{12} = (-9)(5) + (2)(-6) = -45 - 12 = -57
\]

- Element \( C_{21} \):
\[
C_{21} = (8)(-2) + (-1)(3) = -16 - 3 = -19
\]

- Element \( C_{22} \):
\[
C_{22} = (8)(5) + (-1)(-6) = 40 + 6 = 46
\]

- Element \( C_{31} \):
\[
C_{31} = (4)(-2) + (-3)(3) = -8 - 9 = -17
\]

- Element \( C_{32} \):
\[
C_{32} = (4)(5) + (-3)(-6) = 20 + 18 = 38
\]

#### Final Result:
\[
\begin{bmatrix}
24 & -57 \\
-19 & 46 \\
-17 & 38
\end{bmatrix}
\]

---

Problem 4:

\[
\begin{bmatrix}
2 & -1 \\
5 & 8 \\
4 & 0
\end{bmatrix}
*
\begin{bmatrix}
3 & 2 \\
-4 & -7
\end{bmatrix}
\]

#### Step 1: Dimensions Check
- The first matrix is \( 3 \times 2 \).
- The second matrix is \( 2 \times 2 \).
- The result will be a \( 3 \times 2 \) matrix.

#### Step 2: Compute Each Element

- Element \( C_{11} \):
\[
C_{11} = (2)(3) + (-1)(-4) = 6 + 4 = 10
\]

- Element \( C_{12} \):
\[
C_{12} = (2)(2) + (-1)(-7) = 4 + 7 = 11
\]

- Element \( C_{21} \):
\[
C_{21} = (5)(3) + (8)(-4) = 15 - 32 = -17
\]

- Element \( C_{22} \):
\[
C_{22} = (5)(2) + (8)(-7) = 10 - 56 = -46
\]

- Element \( C_{31} \):
\[
C_{31} = (4)(3) + (0)(-4) = 12 + 0 = 12
\]

- Element \( C_{32} \):
\[
C_{32} = (4)(2) + (0)(-7) = 8 + 0 = 8
\]

#### Final Result:
\[
\begin{bmatrix}
10 & 11 \\
-17 & -46 \\
12 & 8
\end{bmatrix}
\]

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Final Answers:

1. \(\boxed{\begin{bmatrix} -31 & 30 \\ 18 & -12 \end{bmatrix}}\)
2. \(\boxed{\begin{bmatrix} 42 & -39 \\ -48 & 51 \end{bmatrix}}\)
3. \(\boxed{\begin{bmatrix} 24 & -57 \\ -19 & 46 \\ -17 & 38 \end{bmatrix}}\)
4. \(\boxed{\begin{bmatrix} 10 & 11 \\ -17 & -46 \\ 12 & 8 \end{bmatrix}}\)