Question

Answer the following question:

Find minor and cofactor of elements of the determinant:

`|(1, -1, 2),(3, 0, -2),(1, 0, 3)|`

Answer 1

Here, `|("a"_11, "a"_12, "a"_13),("a"_21, "a"_22, "a"_23),("a"_31, "a"_32, "a"_33)| = |(1, -1, 2),(3, 0, -2),(1, 0, 3)|`

M₁₁ = `|(0, -2),(0, 3)|` = 0 − 0 = 0

∴ C₁₁ = (– 1)¹⁺¹ M₁₁ = 1(0) = 0

M₁₂ = `|(3, -2),(1, 3)|` = 9 + 2 = 11

∴ C₁₂ = (– 1)¹⁺² M₁₂ = (– 1)(11) = −11

M₁₃ = `|(3, 0),(1, 0)|` = 0 –  0 = 0

∴ C₁₃ = (– 1)¹⁺³ M₁₃ = 1(0) = 0

M₂₁ = `|(-1, 2),(0, 3)|` = −3 − 0 = −3

∴ C₂₁ = (–1)²⁺¹ M₂₁ = (–1)(−3) = 3

M₂₂ = `|(1, 2),(1, 3)|` = 3 – 2 = 1

∴ C₂₂ = (–1)²⁺² M₂₂ = 1(1) = 1

M₂₃ = `|(1, -1),(1, 0)|` = 0 + 1 =  1

∴ C₂₃ = (–1)²⁺³ M₂₃ = (–1)(1) = –1

M₃₁ = `|(-1, 2),(0, -2)|` = 2 – 0 = 2

∴ C₃₁ = (–1)³⁺¹ M₃₁ = 1(2) = 2

M₃₂ = `|(1, 2),(3, -2)|` = –2 − 6 = −8

∴ C₃₂ = (–1)³⁺² M₃₂ = (–1)(−8) = 8

M₃₃ = `|(1, -1),(3, 0)|` = 0 + 3 = 3

∴ C₃₃ = (–1)³⁺³ M₃₃ = 1(3) = 3