Question

Find the co-factor of the element of the following matrix.

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

Answer 1

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

The co-factor of a_ij is given by A_ij = (−1)^i+j M_ij

Now, M₁₁ = `|(3,5),(0,-1)|` = − 3 − 0 = − 3

∴ A₁₁ = (− 1)¹⁺¹(− 3) = − 3

M₁₂ = `|(-2,5),(-2,-1)|` = 2 + 10 = 12

∴ A₁₂ = (− 1)¹⁺²(12) = − 12

M₁₃ = `|(-2,3),(-2,0)|` = 0 + 6 = 6

∴ A₁₃ = (− 1)¹⁺³(6) = 6

M₂₁ = `|(-1,2),(0,-1)|` = 1 − 0 = 1

∴ A₂₁ = (− 1)²⁺¹(1) = − 1

M₂₂ = `|(1,2),(-2,-1)|` = − 1 + 4 = 3

∴ A₂₂ = (− 1)²⁺²(3) = 3

M₂₃ = `|(1,-1),(-2,0)|` = 0 − 2 = − 2

∴ A₂₃ = (−1)²⁺³(− 2) = 2

M₃₁ = `|(-1,2),(3,5)|` = − 5 − 6 = − 11

∴ A₃₁ = (− 1)³⁺¹(− 11) = − 11

M₃₂ = `|(1,2),(-2,5)|` = 5 + 4 = 9

∴ A₃₂ = (− 1)³⁺²(9) = − 9

M₃₃ = `|(1,-1),(-2,3)|` = 3 − 2 = 1

∴ A₃₃ = (− 1)³⁺³(1) = 1