Question

Verify A(adj A) = (adj A)A = |A|I.

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

Answer 1

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

|A| = 1[0 − 0] + 1[9 + 2] + 2[0 − 0]

= 1 × 11

= 11

A₁₁ = `(-1)^(1 + 1) |(0,-2),(0,3)|`

= (−1)² [0 − 0]

= 0

A₁₂ = `(-1)^(1 + 2) |(3,-2),(1,3)|`

= (−1)³ [9 + 2]

= −11

A₁₃ = `(-1)^(1 + 3) |(3,0),(1,0)|`

= (−1)⁴ [0 − 0]

= 0

A₂₁ = `(-1)^(2 + 1) |(-1,2),(0,3)|`

= (−1)³ [−3 − 0]

= −1 × (−3)

= 3

A₂₂ = `(- 1)^(2 + 2) |(1, 2),(1,3)|`

= (−1)⁴ [3 − 2]

= 1 × 1

= 1

A₂₃ = `(-1)^(2+ 3) |(1,-1),(1,0)|`

= (−1)⁵ [0 + 1]

= −1

A₃₁ = `(1)^(3 + 1) |(-1,2),(0,-2)|`

= (−1)⁴ [2 − 0]

= 1 × 2

= 2

A₃₂ = `(-1)^(3 + 2) |(1,2),(3,-2)|`

= (−1)⁵ [−2 − 6]

= −1 × (−8)

= 8

A₃₃ = `(-1)^(3 + 3) |(1,-1),(3,0)|`

= (−1)⁶ [0 + 3]

= 1 × 3

= 3

adj A = `[(0,-11,0),(3,1,-1),(2,8,3)] = [(0,3,2),(-11,1,8),(0,-1,3)]`

L.H.S. = A(adj A) = `[(1,-1,2),(3,0,-2),(1,0,3)] [(0,3,2),(-11,1,8),(0,-1,3)]`

= `[(1 xx 0 + (- 1) xx (- 11) + 2 xx 0, 1 xx 3 + (- 1) xx 1 + 2 xx (- 1), 1 xx 2 + (- 1) xx 8 + 2 xx 3),(3 xx 0 + 0 xx (- 11) + (- 2) xx 0, 3 xx 3 + 0 xx 1 + (- 2) xx (- 1), 3 xx 2 + 0 xx 8 + (- 2) xx 3),(1 xx 0 + 0 xx (- 11) + 3 xx 0, 1 xx 3 + 0 xx 1 + 3 xx (-1), 1 xx 2 + 0 xx 8 + 3 + 3)]`

= `[(0+11+0,3 - 1 - 2, 2 - 8 + 6),(0+0+0, 9 + 0 + 2, 6 + 0 - 6),(0 + 0 + 0, 3 + 0 - 3, 2 + 0 + 9)]`

= `[(11,0,0),(0,11,0),(0,0,11)]`

= `11[(1,0,0),(0,1,0),(0,0,1)]`

= 11 · I

= |A| · I

R.H.S. = (adj A)A `= [(0,3,2),(-11,1,8),(0,-1,3)][(1,-1,2),(3,0,-2),(1,0,3)]`

= `[(0xx3 + 3 xx 3 + 2 xx 1,0xx(-1) + 3 xx 0 + 2 xx 0,0 xx 2 + 3 xx (- 2) + 2 xx 3),(-11xx1 + 1 xx 3 + 8 xx 1, -11 xx (- 1) + 1 xx 0 + 8 xx 0,-11 xx 2 + 1 xx(- 2) + 8 xx3),(0 xx 1 + (- 1) xx 3 + 3 xx 1, 0xx(- 1) + (- 1) xx 0 + 3 xx 0, 0xx2 + (- 1)xx (- 2) + 3 xx 3)]`

`= [(0 + 9 + 2, 0 + 0 + 0, 0 - 6 + 6),(- 11 + 3 + 8, 11 + 0 + 0, - 22 - 2 + 24),(0 - 3 + 3, 0 + 0 + 0, 0 + 2 + 9)]`

= `[(11,0,0),(0,11,0),(0,0,11)]`

= `11 [(1,0,0),(0,1,0),(0,0,1)]`

= 11 · I

= |A| · I

Hence, A(adj A) = (adj A)A = |A|I