For α, β, γ ∈ R, let A=[α2 6 8 3 β2 9 4 5 γ2] and B=[2 α 3 5 2 2 β 6 1 4 2 γ-3]. If trace A= trace B, then the value of (α-1+β-1+γ-1) is equal to

Question

For α, β, &gamma; ∈ R, let A=[α2 6 8 3 β2 9 4 5 &gamma;2] and B=[2 α 3 5 2 2 β 6 1 4 2 &gamma;-3]. If trace A= trace B, then the value of (α-1+β-1+&gamma;-1) is equal to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

We have trace A= traceB, so α2+β2+&gamma;2=2 α+2 β+2 &gamma;-3 ⇒(α2-2 α+1)+(β2-2 β+1)+(&gamma;2-2 &gamma;+1)=0 ⇒(α-1)2+(β-1)2+(&gamma;-1)2=0 ⇒ α=1, β=1, &gamma;=1 Hence, (α-1+β-1+&gamma;-1) =(1+1+1)=3 .

Q. For α,β,γ∈R, let A=⎣⎡​α234​6β25​89γ2​⎦⎤​ and B=⎣⎡​2α21​32β4​562γ−3​⎦⎤​. If trace A= trace B, then the value of (α−1+β−1+γ−1) is equal to

We have trace A= traceB, so α2+β2+γ2=2α+2β+2γ−3 ⇒(α2−2α+1)+(β2−2β+1)+(γ2−2γ+1)=0 ⇒(α−1)2+(β−1)2+(γ−1)2=0 ⇒α=1,β=1,γ=1 Hence, (α−1+β−1+γ−1) =(1+1+1)=3.

Q. For $α, β, γ \in R,$ let $A = ⎣ ⎡ α^{2} 34 6 β^{2} 5 89 γ^{2} ⎦ ⎤$ and $B = ⎣ ⎡ 2 α 21 3 2 β 4 56 2 γ - 3 ⎦ ⎤$ . If trace $A =$ trace $B$ , then the value of $(α^{- 1} + β^{- 1} + γ^{- 1})$ is equal to