Evaluate the determinants:

• (i) | 3 -1 -2 | 0 -1 3 | -5 0 |
• (ii) | 3 -4 5 | 1 -2 2 | 3 1 |
• (iii) | 0 1 2 | -1 0 -3 | -2 3 0 |
• (iv) | 2 -1 -2 | 0 2 -1 | 3 -5 0 |

To evaluate the determinants of the given matrices, we will use the formula for a 3x3 determinant. The determinant of a matrix:

Matrix (i)

For the matrix:

| 3 -1 -2 |
| 0 -1 3 |
| -5 0 0 |

The determinant can be calculated as:

Det = 3((-1)(0) - (3)(0)) - (-1)((0)(0) - (3)(-5)) - 2((0)(0) - (-1)(-5))

Det = 3(0) + 1(15) - 2(0) = 15

Matrix (ii)

For the matrix:

| 3 -4 5 |
| 1 -2 2 |
| 3 1 0 |

Using the determinant formula:

Det = 3((-2)(0) - (2)(1)) - (-4)((1)(0) - (2)(3)) + 5((1)(1) - (-2)(3))

Det = 3(0 - 2) + 4(0 - 6) + 5(1 + 6) = -6 - 24 + 35 = 5

Matrix (iii)

For the matrix:

| 0 1 2 |
| -1 0 -3 |
| -2 3 0 |

Calculating the determinant:

Det = 0((0)(0) - (-3)(3)) - 1((-1)(0) - (-3)(-2)) + 2((-1)(3) - (0)(-2))

Det = 0 + 1(0 - 6) + 2(-3) = -6 - 6 = -12

Matrix (iv)

For the matrix:

| 2 -1 -2 |
| 0 2 -1 |
| 3 -5 0 |

Using the determinant formula:

Det = 2((2)(0) - (-1)(-5)) - (-1)((0)(0) - (-1)(3)) - 2((0)(-5) - (2)(3))

Det = 2(0 - 5) + 1(0 + 3) - 2(0 - 6) = -10 + 3 + 12 = 5

Summary of Determinants

• Determinant of Matrix (i): 15
• Determinant of Matrix (ii): 5
• Determinant of Matrix (iii): -12
• Determinant of Matrix (iv): 5