Question

Evaluate A = `|(2, -3,5),(6, 0, 4),(1, 5, -7)|` Also find minor and cofactor of elements in the 2^nd row of determinant and verify a₂₁ C₂₁ + a₂₂ C₂₂ + a₂₃ C₂₃ = value of A

where M₂₁, M₂₂, M₂₃ are minors of a₂₁, a₂₂, a₂₃ and

C₂₁, C₂₂, C₂₃ are cofactors of a₂₁, a₂₂, a₂₃.

Answer 1

Let A = `|("a"_11, "a"_12, "a"_13),("a"_21, "a"_22, "a"_23),("a"_31, "a"_32, "a"_33)| = |(2, -3, 5),(6, 0, 4),(1, 5, -7)|`

∴ A = `2|(0, 4),(5, -7)| -(-3)|(6, 4),(1, -7)| + 5|(6, 0),(1, 5)|`

= 2 (0 – 20) + 3 ( –42 – 4) + 5 (30 – 0)

= 2 ( – 20) + 3 ( – 46) + 5 (30)

= –40 – 138 + 150

= – 28.     ...(1)

Also, a₂₁ = 6, a₂₂ = 0, a₂₃ = 4

∴ M₂₁ = `|(-3, 5),(5, -7)|` = 21 – 25 = – 4

∴ C₂₁ = (– 1)²⁺¹ M₂₁ = – 1(– 4) = 4

M₂₂ = `|(2, 5),(1, -7)|` = – 14 – 5 = – 19

∴ C₂₂ = (– 1)²⁺² M₂₂ = 1(– 19) = – 19

M₂₃ = `|(2, -3),(1, 5)|` = 10 – ( – 3) = 13

∴ C₂₃ = (– 1)²⁺³ M₂₃ = – 1(13) = – 13

a₂₁ C₂₁ + a₂₂ C₂₂ + a₂₃ C₂₃ 

= 6(4) + 0(–19) + 4(– 13)

= 24 – 0 – 52

= – 28

= value of A    ...[By (1)]