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

Sure! Let’s solve each matrix multiplication problem step by step. Remember: to multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The result will be a matrix with dimensions (rows of first) × (columns of second).

We’ll use the standard rule:
> To find element at row *i*, column *j* in the product matrix, take the dot product of row *i* from the first matrix and column *j* from the second matrix.

---

Problem 1:

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

This is a 2×2 times 2×2 → result is 2×2.

Element (1,1): \(3*(-2) + (-5)*5 = -6 -25 = -31\)
Element (1,2): \(3*0 + (-5)*(-6) = 0 + 30 = 30\)
Element (2,1): \((-4)*(-2) + 2*5 = 8 + 10 = 18\)
Element (2,2): \((-4)*0 + 2*(-6) = 0 -12 = -12\)

✔ Answer 1:
\[
\begin{bmatrix}
-31 & 30 \\
18 & -12
\end{bmatrix}
\]

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Problem 2:

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

Again, 2×2 × 2×2 → 2×2.

Element (1,1): \((-2)*(-7) + 7*4 = 14 + 28 = 42\)
Element (1,2): \((-2)*9 + 7*(-3) = -18 -21 = -39\)
Element (2,1): \(4*(-7) + (-5)*4 = -28 -20 = -48\)
Element (2,2): \(4*9 + (-5)*(-3) = 36 + 15 = 51\)

✔ Answer 2:
\[
\begin{bmatrix}
42 & -39 \\
-48 & 51
\end{bmatrix}
\]

---

Problem 3:

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

This is 3×2 × 2×2 → result is 3×2.

Row 1:
- (1,1): \((-9)*(-2) + 2*3 = 18 + 6 = 24\)
- (1,2): \((-9)*5 + 2*(-6) = -45 -12 = -57\)

Row 2:
- (2,1): \(8*(-2) + (-1)*3 = -16 -3 = -19\)
- (2,2): \(8*5 + (-1)*(-6) = 40 + 6 = 46\)

Row 3:
- (3,1): \(4*(-2) + (-3)*3 = -8 -9 = -17\)
- (3,2): \(4*5 + (-3)*(-6) = 20 + 18 = 38\)

✔ Answer 3:
\[
\begin{bmatrix}
24 & -57 \\
-19 & 46 \\
-17 & 38
\end{bmatrix}
\]

---

Problem 4:

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

Again, 3×2 × 2×2 → 3×2.

Row 1:
- (1,1): \(2*3 + (-1)*(-4) = 6 + 4 = 10\)
- (1,2): \(2*2 + (-1)*(-7) = 4 + 7 = 11\)

Row 2:
- (2,1): \(5*3 + 8*(-4) = 15 - 32 = -17\)
- (2,2): \(5*2 + 8*(-7) = 10 - 56 = -46\)

Row 3:
- (3,1): \(4*3 + 0*(-4) = 12 + 0 = 12\)
- (3,2): \(4*2 + 0*(-7) = 8 + 0 = 8\)

✔ Answer 4:
\[
\begin{bmatrix}
10 & 11 \\
-17 & -46 \\
12 & 8
\end{bmatrix}
\]

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## ✔ Final Answers:

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

Let me know if you’d like to see any step visualized differently or need help understanding the dot product process!