Matrices and matrix operations worksheet with six problems.

Let’s solve each problem one by one, carefully and step by step.

---

Problem 1: Find the dimensions of matrix X

Matrix X is:

```
X = [ -4 5 -5 2 1 ]
[ -4 -8 -2 0 -6 ]
```

Dimensions are written as rows × columns.

- How many rows? → 2 rows (top row and bottom row)
- How many columns? → 5 columns (each number in a row is a column)

So dimensions = 2 by 5

✔ Correct answer: 2 by 5

---

Problem 2: Find the value of b₂₁

Matrix B is:

```
B = [ -7 8 -2 -3 -7 ]
[ -9 6 -2 -7 -2 ]
[ -6 10 7 -6 1 ]
```

The subscript “b₂₁” means: row 2, column 1

Look at row 2: [-9, 6, -2, -7, -2]

Column 1 of that row is the first number → -9

✔ Answer: -9

---

Problem 3: Find the value of p₁₄

We’re told:

P = first matrix + second matrix

First matrix:
```
[ 1 -10 -10 2 -6 ]
[ 3 -2 2 -3 7 ]
```

Second matrix:
```
[ -9 -1 -9 -2 -2 ]
[ -2 2 -10 9 9 ]
```

Add them element by element.

We need p₁₄ → row 1, column 4

In row 1, column 4:

First matrix has: 2

Second matrix has: -2

Add: 2 + (-2) = 0

✔ Answer: 0

---

Problem 4: Find the value of r₁₃

R = first matrix minus second matrix

First matrix:
```
[ 10 -3 4 ]
[ -2 -3 -4 ]
[ 3 6 4 ]
[ 3 6 -10 ]
```

Second matrix:
```
[ 1 -9 2 ]
[-10 -6 -9 ]
[ -9 3 -7 ]
[ -3 -8 -10 ]
```

We need r₁₃ → row 1, column 3

Row 1, col 3 of first matrix: 4

Row 1, col 3 of second matrix: 2

Subtract: 4 - 2 = 2

✔ Answer: 2

---

Problem 5: Find the value of s₁₁

S = 7 × matrix

Matrix inside:
```
[ 1 -7 -1 6 -1 ]
[ 5 4 10 -8 2 ]
```

s₁₁ → row 1, column 1 of the result

Original element at row 1, col 1 is 1

Multiply by 7: 7 × 1 = 7

✔ Answer: 7

---

Problem 6: Find the value of c₂₁

C = -10 × first matrix + 3 × second matrix

First matrix:
```
[ 2 -4 3 ]
[ 5 7 5 ]
```

Second matrix:
```
[ -3 0 -2 ]
[ 0 -3 7 ]
```

We need c₂₁ → row 2, column 1 of final result

Step 1: Multiply first matrix by -10

Element at row 2, col 1: 5 × (-10) = -50

Step 2: Multiply second matrix by 3

Element at row 2, col 1: 0 × 3 = 0

Step 3: Add them: -50 + 0 = -50

✔ Answer: -50

---

Final Answer:
1. 2 by 5
2. -9
3. 0
4. 2
5. 7
6. -50