If A and B are non-singular matrices of order three such that adj(A B)=[ 1 1 1 1 α 1 1 1 α ] and |B2 a d j A|=α 2+3α -8 , then the value of α is equal to

Question

If A and B are non-singular matrices of order three such that adj(A B)=[ 1 1 1 1 α 1 1 1 α ] and |B2 a d j A|=α 2+3α -8 , then the value of α is equal to (A) (9/5) (B) (8/5) (C) 3 (D) 2

Tardigrade Admin · Accepted Answer

|(a d j A B)|=| 1 1 1 1 α 1 1 1 α | =1((α )2 - 1)-1(α - 1)+1(1 - α ) =(α )2-2α +1=(α - 1)2 ⇒ (|A B|)2=(α - 1)2 and |B2 a d j A|=|B|2|A|2 =(|A B|)2=(α - 1)2=(α )2+3α -8 ⇒ α 2-2α +1=α 2+3α -8 ⇒ 5α =9 ⇒ α =(9/5)

Q. If $A$ and $B$ are non-singular matrices of order three such that $a d j (A B) = ⎣ ⎡ 111 1 α 1 11 α ⎦ ⎤$ and $∣ ∣ B^{2} a d j A ∣ ∣ = α^{2} + 3 α - 8$ , then the value of $α$ is equal to

$\frac{9}{5}$

$\frac{8}{5}$

$3$

$2$

Q. If A and B are non-singular matrices of order three such that adj(AB)=⎣⎡​111​1α1​11α​⎦⎤​ and ∣∣​B2adjA∣∣​=α2+3α−8 , then the value of α is equal to

59​

58​

3

2

∣(adjAB)∣=∣∣​111​1α1​11α​∣∣​ =1((α)2−1)−1(α−1)+1(1−α) =(α)2−2α+1=(α−1)2 ⇒(∣AB∣)2=(α−1)2 and ∣∣​B2adjA∣∣​=∣B∣2∣A∣2 =(∣AB∣)2=(α−1)2=(α)2+3α−8 ⇒α2−2α+1=α2+3α−8 ⇒5α=9 ⇒α=59​

Q. If $A$ and $B$ are non-singular matrices of order three such that $a d j (A B) = ⎣ ⎡ 111 1 α 1 11 α ⎦ ⎤$ and $∣ ∣ B^{2} a d j A ∣ ∣ = α^{2} + 3 α - 8$ , then the value of $α$ is equal to

$\frac{9}{5}$

$\frac{8}{5}$

$3$

$2$