If X is a non-zero column matrix, such that A X=λ X where λ is a scalar and the matrix A is [4 6 6 1 3 2 -1 -5 -2], then sum of distinct values of λ is

Question

If X is a non-zero column matrix, such that A X=λ X where λ is a scalar and the matrix A is [4 6 6 1 3 2 -1 -5 -2], then sum of distinct values of λ is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

A X=λ X ⇒(A-λ I) X=0 but X ≠ 0 ⇒|A-λ I|=0 ⇒ |4-λ 6 6 1 3-λ 2 -1 -5 -2-λ|=0 ⇒(λ-1)(λ-2)2=0 ⇒ λ=1,2,2 Sum of distinct values 1+2=3

Q. If $X$ is a non-zero column matrix, such that $A X = λ X$ where $λ$ is a scalar and the matrix $A$ is $⎣ ⎡ 41 - 1 63 - 5 62 - 2 ⎦ ⎤$ , then sum of distinct values of $λ$ is

$A X = λ X \Rightarrow (A - λ I) X = 0$
but $X \neq = 0 \Rightarrow ∣ A - λ I ∣ = 0$
$\Rightarrow ∣ ∣ 4 - λ 1 - 1 6 3 - λ - 5 62 - 2 - λ ∣ ∣ = 0$
$\Rightarrow (λ - 1) (λ - 2)^{2} = 0$
$\Rightarrow λ = 1, 2, 2$
Sum of distinct values $1 + 2 = 3$

Q. If X is a non-zero column matrix, such that AX=λX where λ is a scalar and the matrix A is ⎣⎡​41−1​63−5​62−2​⎦⎤​, then sum of distinct values of λ is

AX=λX⇒(A−λI)X=0 but X=0⇒∣A−λI∣=0 ⇒∣∣​4−λ1−1​63−λ−5​62−2−λ​∣∣​=0 ⇒(λ−1)(λ−2)2=0 ⇒λ=1,2,2 Sum of distinct values 1+2=3

Q. If $X$ is a non-zero column matrix, such that $A X = λ X$ where $λ$ is a scalar and the matrix $A$ is $⎣ ⎡ 41 - 1 63 - 5 62 - 2 ⎦ ⎤$ , then sum of distinct values of $λ$ is

$A X = λ X \Rightarrow (A - λ I) X = 0$
but $X \neq = 0 \Rightarrow ∣ A - λ I ∣ = 0$
$\Rightarrow ∣ ∣ 4 - λ 1 - 1 6 3 - λ - 5 62 - 2 - λ ∣ ∣ = 0$
$\Rightarrow (λ - 1) (λ - 2)^{2} = 0$
$\Rightarrow λ = 1, 2, 2$
Sum of distinct values $1 + 2 = 3$