Question

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

verify that A (adj A) = (adj A) A = |A| . I

Answer 1

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

`therefore |A| = |(1,-1,2),(3,0,-2),(1,0,3)|`

`= 1 xx |(0,-2),(0,3)| - (-1) xx |(3,-2),(1,3)| + 2 xx |(3,0),(1,0)|`

= 1 (0 - 0) + (9 + 2) + 2 (0 - OJ = 0 + I + O 

= 11

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

A₁₁ = (-1)¹⁺¹ M₁₁ 

       = (-1)² (0) = 0

A₁₂ = (-1)¹⁺² M₁₂ = (-1)³ (9 + 2) = -11

A₁₃ = (-1)¹⁺³ M₁₃ = (-1)⁴ (0) = 0

A₂₁ = (-1)²⁺¹ M₂₁ = (-1)³ (-3-0) = 3

A₂₂ = (-1)²⁺² M₂₂ = (-1)⁴ (3 -2) = 1

A₂₃ = (-1)²⁺³ M₂₃ = (-1)⁵ (0 + 1) = -1

A₃₁ = (-1)^{3 + 1} M₃₁ = (-1)⁴ (2 - 0) = 2

A₃₂ = (-1)³⁺² M₃₂ = (-1)⁵ (-2 - 8) = 8

A₃₃ = (-1)³⁺³ M₃₃ = (-1)⁶ (0 + 3) = 3 

∴ Malrlx of cofactor Is `[("A"_11 , "A"_12 , "A"_13),("A"_21,"A"_22 ,"A"_23),("A"_31,"A"_32 ,"A"_33)]`

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

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

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

`= |(11,0,0),(0,11,0),(0,0,11)|`   .......(I)

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

`= |(11,0,0),(0,11,0),(0,0,11)|`   .......(II)

|A| . I = `11 [(1,0,0),(0,1,0),(0,0,1)]`

`= |(11,0,0),(0,11,0),(0,0,11)|`   .......(III)

From (I)(, (II) and (III). we get 

A (adj A) =(adj A) A = |A| I