Question

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

Answer 1

The co-factor A_ij of a_ij is equal to (– 1)^i+j M_ij.
Here,
a₁₁ = 1
∴ M₁₁ = – 8
and A₁₁ = (– 1)¹⁺² M₁₁ = (1) (– 8) = – 8
a₁₂ = 2
∴ M₁₁ = 5
and A₁₂ = (– 1)¹⁺² M₁₂ = (– 1) (5) = – 5
a₂₁ = 5
∴ M₂₁ = 2
and A₂₁ = (– 1)²⁺¹ M₂₁ = (– 1) (2) = – 2
a₂₂ = – 8
∴ M₂₂ = 1
and A₂₂ = (– 1)²⁺² M₂₂ = (1) (1) = 1
∴ The matrix of the co-factors is

[A_ij]_2x2 = `[("A"_11, "A"_12),("A"_21, "A"_22)]`

= `[(-8, -5),(-2, 1)]`.