Question

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

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

Answer 1

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

Here, a₁₁ = 1

∴ M₁₁ = `|(1,3),(3,-5)|` =  −5 − 9 = −14

and A₁₁ = (−1)¹⁺¹ (−14) = −14

a₁₂ = 0

∴ M₁₂ = `|(-2,3),(0,-5)|` = 10 − 0 = 10

and A₁₂ = (−1)¹⁺² (10) = −10

a₁₃ = 2

∴ M₁₃ = `|(-2,1),(0,3)|` = −6 − 0 = −6

and A₁₃ = (−1)¹⁺³ (−6) = −6

a₂₁ = −2

∴ M₂₁ = `|(0,2),(3,-5)|` = 0 − 6 = −6

and A₂₁ = (−1)²⁺¹ (−6) = 6

a₂₂ = 1

∴ M₂₂ = `|(1,2),(0,-5)|` = −5 − 0 = −5

and A₂₂ = (−1)²⁺² (−5) = −5

a₂₃ = 3

∴  M₂₃ = `|(1,0),(0,3)|` = 3 − 0 = 3

and A₂₃ = (−1)²⁺³ (3) = −3

a₃₁ = 0

∴ M₃₁ = `|(0,2),(1,3)|` = 0 − 2 = −2

and A₃₁ = (−1)³⁺¹ (−2) = −2

a₃₂ = 3

∴ M₃₂ = `|(1,2),(-2,3)|` = 3 + 4 = 7

and A₃₂ = (−1)³⁺² (7) = −7

a₃₃ = −5

∴ M₃₃ = `|(1,0),(-2,1)|` = 1 − 0 = 1

and A₃₃ = (−1)³⁺³ (1) = 1

∴ The matrix of the co-factor is

`[("A"_11, "A"_12, "A"_13),("A"_21, "A"_22, "A"_23),("A"_31, "A"_32, "A"_33)]` = `[(-14,-10,-6),(6,-5,-3),(-2,-7,1)]`