Question

Find the adjoint of the matrices.

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

Answer 1

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

Let C_ij be cofactors of a_ij in A.

C₁₁ = `(-1)^(1 + 1) |(3,5),(0,1)|`

= 3 − 0

= 3

C₁₂ = `(-1)^(1 + 2) |(2,5),(-2,1)|`

= −(2 + 10)

= −12

C₁₃ = `(-1)^(1 + 3) |(2,3),(-2,0)|`

= 0 + 6

= 6

C₂₁ = `(-1)^(2 + 1) |(-1,2),(0,1)|`

= −(−1 − 0)

= 1

C₂₂ = `(-1)^(2 + 2) |(1,2),(-2,1)|`

= 1 + 4

= 5

C₂₃ = `(-1)^(2 + 3) |(1,-1),(-2,0)|`

= −(0 − 2)

= 2

C₃₁ = `(-1)^(3 + 1) |(-1,2),(3,5)|`

= −5 − 6

= −11

C₃₂ = `(-1)^(3 + 2) |(1,2),(2,5)|`

= −(5 − 4)

= −l

C₃₃ = `(-1)^(3 + 3) |(1,-1),(2,3)|`

= 3 + 2

= 5

∴ Adj A `=  [(C_11,C_12,C_13),(C_21,C_22,C_23),(C_31,C_32,C_33)]^T`

= `[(3,-12,6),(1,5,2),(-11,-1,5)]^T`

= `[(3,1,-11),(-12,5,-1),(6,2,5)]`