Question

Find the minor and cofactor of element of the determinant

D = `|(2, -1, 3),(1, 2, -1),(5, 7, 2)|`

Answer 1

Here,

Let D = `|("a"_11, "a"_12, "a"_13),("a"_21, "a"_22, "a"_23),("a"_31, "a"_32, "a"_33)| = |(2, -1, 3),(1, 2, -1),(5, 7, 2)|`

M₁₁ = `|(2, -1),(7, 2)|` = 4 + 7 = 11

C₁₁ = (– 1)¹⁺¹ M₁₁ = (-1)² (11) = (1)(11) = 11

M₁₂ = `|(1, -1),(5, 2)|` = 2 - (- 5) = 2 + 5 = 7

C₁₂ = (– 1)¹⁺² M₁₂ = (– 1)³ (7) = – 7

M₁₃ = `|(1, 2),(5, 7)|` = 7 – 10 = –  3

C₁₃ = (– 1)¹⁺³ M₁₃ = (1)(– 3) = – 3

M₂₁ = `|(-1, 3),(7, 2)|` = –2 – 21 = – 23

C₂₁ = (– 1)²⁺¹ M₂₁ = (– 1)(– 23) = 23

M₂₂ = `|(2, 3),(5, 2)|` = 4 – 15 = – 11

C₂₂ = (– 1)²⁺² M₂₂ = (1)(– 11) = – 11

M₂₃ = `|(2, -1),(5, 7)|` = 14 - (- 5) = 14 + 5 = 19

C₂₃ = (– 1)²⁺³ M₂₃ = (– 1)(19) = – 19

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

C₃₁ = (– 1)³⁺¹ M₃₁ = (1)(– 5) = – 5

M₃₂ = `|(2, 3),(1, -1)|` = –2 – 3 = – 5

C₃₂ = (– 1)³⁺² M₃₂ = (– 1)(– 5) = 5

M₃₃ = `|(2,-1),(1, 2)|` = 4 - (- 1) = 4 + 1 = 5

C₃₃ = (– 1)³⁺³ M₃₃ = (- 1)⁶ (5) = 5