Question

Find the adjoint 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)]`

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

The co-factor matrix = `[("A"_11, "A"_12, "A"_13),("A"_21, "A"_22, "A"_23),("A"_31, "A"_32, "A"_33)]`

∴ adj A = `[(-3, -12, 6),(-1, 3, 2),(-11, -9, 1)]^"T"`

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