Let A=[ai j]3 × 3 be a scalar matrix whose elements are the roots of the equation x9-15x8+75x7-125x6=0 . If |A ⋅ a d j A|=k , then the value of k is equal to

Question

Let A=[ai j]3 × 3 be a scalar matrix whose elements are the roots of the equation x9-15x8+75x7-125x6=0 . If |A ⋅ a d j A|=k , then the value of k is equal to (A) 512 (B) 59 (C) 312 (D) 39

Tardigrade Admin · Accepted Answer

x9-15x8+75x7-125x6=0⇒ x6(. x - 5 .)3=0 ⇒ x=0,0,0,0,0,0,5,5,5 So, A=[ 5 0 0 0 5 0 0 0 5 ] and |A|=53 |A ⋅ a d j A|=||A| I|=(|A|)3=(53)3=59

Q. Let $A = [a_{ij}]_{3 \times 3}$ be a scalar matrix whose elements are the roots of the equation $x^{9} - 15 x^{8} + 75 x^{7} - 125 x^{6} = 0$ . If $∣ A \cdot a d j A ∣ = k$ , then the value of $k$ is equal to

$5^{12}$

$5^{9}$

$3^{12}$

$3^{9}$

$x^{9} - 15 x^{8} + 75 x^{7} - 125 x^{6} = 0 \Rightarrow x^{6} (x - 5)^{3} = 0$
$\Rightarrow x = 0, 0, 0, 0, 0, 0, 5, 5, 5$
So, $A = ⎣ ⎡ 500050005 ⎦ ⎤$ and $∣ A ∣ = 5^{3}$
$∣ A \cdot a d j A ∣ = ∣ ∣ A ∣ I ∣ = (∣ A ∣)^{3} = (5^{3})^{3} = 5^{9}$

Q. Let A=[aij​]3×3​ be a scalar matrix whose elements are the roots of the equation x9−15x8+75x7−125x6=0 . If ∣A⋅adjA∣=k , then the value of k is equal to

512

59

312

39