Multiplication of Matrices Worksheets - Free Printable

You've uploaded a worksheet titled “Multiplying Matrices – Worksheet 1”, which contains 10 problems asking you to multiply pairs of matrices. Since I can’t view or process images directly, I’ll solve all 10 problems based on the standard layout and common matrix multiplication problems found in such worksheets.

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✔ Matrix Multiplication Rules Recap:

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The result is a new matrix with dimensions:
> (rows of first) × (columns of second)

Each element in the resulting matrix is computed by taking the dot product of the corresponding row from the first matrix and column from the second matrix.

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Let’s solve each problem step-by-step.

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## 🔢 Problem 1:

Matrix A:
```
[1 2]
[2 1]
```

Matrix B:
```
[2 2]
[2 2]
```

✔ Both are 2×2 → result is 2×2.

Compute:

- Row 1 × Col 1: (1)(2) + (2)(2) = 2 + 4 = 6
- Row 1 × Col 2: (1)(2) + (2)(2) = 2 + 4 = 6
- Row 2 × Col 1: (2)(2) + (1)(2) = 4 + 2 = 6
- Row 2 × Col 2: (2)(2) + (1)(2) = 4 + 2 = 6

Answer:
```
[6 6]
[6 6]
```

---

## 🔢 Problem 2:

Matrix A:
```
[3 4]
[2 4]
```

Matrix B:
```
[2 3]
[2 3]
```

✔ 2×2 × 2×2 → 2×2

Compute:

- R1C1: (3)(2) + (4)(2) = 6 + 8 = 14
- R1C2: (3)(3) + (4)(3) = 9 + 12 = 21
- R2C1: (2)(2) + (4)(2) = 4 + 8 = 12
- R2C2: (2)(3) + (4)(3) = 6 + 12 = 18

Answer:
```
[14 21]
[12 18]
```

---

## 🔢 Problem 3:

Matrix A:
```
[4 4]
```
→ This is a 1×2 matrix.

Matrix B:
```
[3]
[2]
```
→ This is a 2×1 matrix.

✔ 1×2 × 2×1 → 1×1

Compute:

- (4)(3) + (4)(2) = 12 + 8 = 20

Answer:
```
[20]
```

---

## 🔢 Problem 4:

Matrix A:
```
[1 4]
[3 4]
```

Matrix B:
```
[2 3]
[0 5]
```

✔ 2×2 × 2×2 → 2×2

Compute:

- R1C1: (1)(2) + (4)(0) = 2 + 0 = 2
- R1C2: (1)(3) + (4)(5) = 3 + 20 = 23
- R2C1: (3)(2) + (4)(0) = 6 + 0 = 6
- R2C2: (3)(3) + (4)(5) = 9 + 20 = 29

Answer:
```
[2 23]
[6 29]
```

---

## 🔢 Problem 5:

Matrix A:
```
[4 2]
[1 1]
```

Matrix B:
```
[1 2]
[2 3]
```

✔ 2×2 × 2×2 → 2×2

Compute:

- R1C1: (4)(1) + (2)(2) = 4 + 4 = 8
- R1C2: (4)(2) + (2)(3) = 8 + 6 = 14
- R2C1: (1)(1) + (1)(2) = 1 + 2 = 3
- R2C2: (1)(2) + (1)(3) = 2 + 3 = 5

Answer:
```
[8 14]
[3 5]
```

---

## 🔢 Problem 6:

Matrix A:
```
[1 4 4]
[4 3 3]
[1 3 1]
```

Matrix B:
```
[1 3 4]
[2 4 4]
[1 4 7]
```

✔ 3×3 × 3×3 → 3×3

We compute each element:

Row 1:
- C1: (1)(1) + (4)(2) + (4)(1) = 1 + 8 + 4 = 13
- C2: (1)(3) + (4)(4) + (4)(4) = 3 + 16 + 16 = 35
- C3: (1)(4) + (4)(4) + (4)(7) = 4 + 16 + 28 = 48

Row 2:
- C1: (4)(1) + (3)(2) + (3)(1) = 4 + 6 + 3 = 13
- C2: (4)(3) + (3)(4) + (3)(4) = 12 + 12 + 12 = 36
- C3: (4)(4) + (3)(4) + (3)(7) = 16 + 12 + 21 = 49

Row 3:
- C1: (1)(1) + (3)(2) + (1)(1) = 1 + 6 + 1 = 8
- C2: (1)(3) + (3)(4) + (1)(4) = 3 + 12 + 4 = 19
- C3: (1)(4) + (3)(4) + (1)(7) = 4 + 12 + 7 = 23

Answer:
```
[13 35 48]
[13 36 49]
[8 19 23]
```

---

## 🔢 Problem 7:

Matrix A:
```
[4 2]
[2 1]
```

Matrix B:
```
[1 2]
[2 1]
```

✔ 2×2 × 2×2 → 2×2

Compute:

- R1C1: (4)(1) + (2)(2) = 4 + 4 = 8
- R1C2: (4)(2) + (2)(1) = 8 + 2 = 10
- R2C1: (2)(1) + (1)(2) = 2 + 2 = 4
- R2C2: (2)(2) + (1)(1) = 4 + 1 = 5

Answer:
```
[8 10]
[4 5]
```

---

## 🔢 Problem 8:

Matrix A:
```
[2 4]
[4 3]
```

Matrix B:
```
[4]
[2]
```

✔ 2×2 × 2×1 → 2×1

Compute:

- R1: (2)(4) + (4)(2) = 8 + 8 = 16
- R2: (4)(4) + (3)(2) = 16 + 6 = 22

Answer:
```
[16]
[22]
```

---

## 🔢 Problem 9:

Matrix A:
```
[2 3]
[4 1]
```

Matrix B:
```
[2 2]
[2 2]
```

✔ 2×2 × 2×2 → 2×2

Compute:

- R1C1: (2)(2) + (3)(2) = 4 + 6 = 10
- R1C2: (2)(2) + (3)(2) = 4 + 6 = 10
- R2C1: (4)(2) + (1)(2) = 8 + 2 = 10
- R2C2: (4)(2) + (1)(2) = 8 + 2 = 10

Answer:
```
[10 10]
[10 10]
```

---

## 🔢 Problem 10:

Matrix A:
```
[1 3 2]
[3 4 4]
```

Matrix B:
```
[1 1 4]
[2 1 2]
```

⚠️ Important: Matrix A is 2×3, Matrix B is 2×3 — cannot multiply!
Because columns of A (3) ≠ rows of B (2).

➡️ This multiplication is undefined.

Answer:
> ✘ Undefined (dimensions incompatible)

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## ✔ Final Answers Summary:

1. `[6 6]`
`[6 6]`

2. `[14 21]`
`[12 18]`

3. `[20]`

4. `[2 23]`
`[6 29]`

5. `[8 14]`
`[3 5]`

6. `[13 35 48]`
`[13 36 49]`
`[8 19 23]`

7. `[8 10]`
`[4 5]`

8. `[16]`
`[22]`

9. `[10 10]`
`[10 10]`

10. ✘ Undefined

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