Question

Find the adjoint of the following matrices : `[(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]`

Answer 1

Let A = `[(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]`

Here,
a₁₁ = 1
∴ M₁₁ = `|(3, 5),(0, -1)|` = – 3 – 0 = – 3
and A₁₁ = (– 1)¹⁺¹ (– 3) = – 3
a₁₂ = – 1

∴ M₁₂ = `|(-2, 5),(-2, -1)|` = 2 + 10 = 12
and A₁₂ = (– 1)¹⁺² (12) = – 12
a₁₃ = 2

∴ M₁₃ = `|(-2, 3),(-2, 0)|` = 0 + 6 = 6
and A₁₃ = (– 1)¹⁺³ (6) = 6
a₂₁ = – 2

∴ M₂₁ = `|(-1, 2),(0, -1)|` = 1 – 0 = 1
and A₂₁ = (– 1)²⁺¹ (1) = – 1
a₂₂ = 3

∴ M₂₂ = `|(1, 2),(-2, -1)|` = – 1 + 4 = 3
and A₂₂ = (– 1)²⁺² (3) = 3
a₂₃ = 5

∴ M₂₃ = `|(1, -1),(-2, 0)|` = 0 – 2 = – 2
and A₂₃ = (– 1)²⁺³ (– 2) = 2
a₃₁ = – 2

∴ M₃₁ = `|(-1, 2),(3, 5)|` = –5 – 6 = – 11
and A₃₁ = (– 1)³⁺¹ (– 11) = – 11
a₃₂ = 0

∴ M₃₂ = `|(1, 2),(-2, 5)|` = 5 + 4 = 9
and A₃₂ = (– 1)³⁺² (9) = – 9
a₃₃ = – 1

∴ M₃₃ = `|(1, -1),(-2, 3)|` = 3 – 2 = 1
and A₃₃ = (– 1)³⁺³ (1) = 1
∴ The matrix of the co-factors is

[A_ij]_3x3 = `[("A"_11, "A"_12, "A"_13),("A"_21, "A"_22, "A"_23),("A"_31, "A"_32, "A"_33)]`

= `[(3, -12, 6),(-1, 3, 2),(-11, -9, 1)]`

Now, adj A = `["A"_"ij"]_(3xx3)^"T" = [(-3, -1, -11),(-12, 3, -9),(6, 2, 1)]`.