Question

Find the cofactor matrix, of the following matrices: `[(5, 8, 7),(-1, -2, 1),(-2, 1, 1)]`

Answer 1

The co-factor Aij of aij is equal to (– 1)i+j Mij.
Here, a₁₁ = 5

∴ M₁₁ = `|(-2, 1),(1, 1)|` = – 2 – 1 = – 3
and A₁₁ = (– 1)¹⁺¹ M₁₁ = (1) (– 3) = – 3
a₁₂ = 8

∴ M₁₂ = `|(-1, 1),(-2, 1)|` = – 1  + 2 = 1
and A₁₂ = (– 1)¹⁺² M₁₁ = (–1) (1) = – 1
a₁₃ = 7

∴ M₁₃ = `|(-1, -2),(-2, 1)|` = – 1  – 4 = – 5
and A₁₃ = (– 1)¹⁺³ M₁₃ = (1) (– 5) = – 5
a₂₁ = – 1

∴ M₂₁ = `|(8, 7),(1, 1)|` = 8  – 7 = 1
and A₂₁ = (– 1)²⁺¹ M₂₁ = (– 1) (1) = – 1
a₂₂ = – 2

∴ M₂₂ = `|(5, 7),(-2, 1)|` = 5  + 14 = 19
and A₂₂ = (– 1)²⁺² M₂₂ = (– 1) (19) = 19
a₂₃ = 1

∴ M₂₃ = `|(5, 8),(-2, 1)|` = 5  + 16 = 21
and A₂₃ = (– 1)²⁺³ M₂₃ = (– 1) (21) = – 21
a₃₁ = – 2

∴ M₃₁ = `|(8, 7),(-2, 1)|` = 8  + 14 = 22
and A₃₁ = (– 1)³⁺² M₃₁ = (1) (22) = 22
a₃₂ = 1

∴ M₃₂ = `|(5, 7),(-1, 1)|` = 5  + 7 = 12
and A₃₂ = (– 1)³⁺² M₃₂ = (– 1) (12) = – 12
a₃₃ = 1

∴ M₃₃ = `|(5, 8),(-1, -2)|` = – 10 + 8 = – 2
and A₃₃ = (– 1)³⁺³ M₃₃ = (1) (– 2) = – 2

∴ 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, -1, -5),(-1, 19, -21),(22, -12, -2)]`.