Matrix Multiplication Algebra 2 Worksheet Printable - Free Printable

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 resulting matrix will have dimensions (rows of first × columns of second).

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Problem 1:

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

This is a 2×2 × 2×2 → result is 2×2.

Compute each element:

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

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

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Problem 2:

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

Again, 2×2 × 2×2 → 2×2.

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

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

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Problem 3:

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

This is a 3×2 × 2×2 → result is 3×2.

Compute each row:

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

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

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

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

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Problem 4:

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

Again, 3×2 × 2×2 → 3×2.

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

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

- Row 3:
- Col 1: \( (4)(3) + (0)(-4) = 12 + 0 = 12 \)
- Col 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 a visual diagram or explanation of how matrix multiplication works conceptually!