Question

Using properties of determinants, prove that: 

`|[a^2 + 1, ab, ac], [ba, b^2 + 1, bc ], [ca, cb, c^2+1]| = a^2 + b^2 + c^2 + 1`

Answer 1

L.H.S.  Δ = `|[a^2 + 1, ab, ac], [ba, b^2 + 1, bc ], [ca, cb, c^2+1]| `

Operating `R_1 → (1)/(a) R_1,R_2 → (1)/(b) R_2 and R_3 → 1/c R_3  "we  have"`

Δ = abc `|[a+ (1)/(a), b, c], [a , b +(1)/(b), c], [a , b , c + (1)/(c)]|`

Multiplying C₁ by a, C₂ by b and C₃ by c, we have

Δ = `|[ a^2+1, -b^2 , c^2], [ a^2, b^2+1, c^2], [a^2, b^2, c^2+1]|`

Operating C₁ → C₁ + C₂+C₃, we have

Δ = `|[ 1+a^2 +b^2+c^2, b^2, c^2],[1+a^2 +b^2+c^2 , b^2+1, c^2],[1+a^2 +b^2+c^2, b^2, c^2+1]|`

Δ = `(1 + a^2 + b^2+c^2) |[ 1, b^2, c^2],[1, b^2+1, c^2], [1, b^2, c^2+1]|` 

Operating R₂ → R₂ → R₁ and R₃ → R₃ → R₁, we have

Δ = `(1 + a^2 + b^2+c^2) |(1,b^2,c^2),(0,1,0),(0,0,1)|`

Expanding along C₁ , we have

Δ = `(1 + a^2 + b^2+c^2) |[1, 0],[0,1]|`

   = `a^2  +b^2+c^2 +1 = R.H.S.`