Question
Let β be a real number. Consider the matrix A = β 0 1 2 1 - 2 3 1 - 2 . If A 7 - β - 1 A 6 - β A 5 is a singular matrix, then the value of 9 β is _____ .
JEE Advanced
Mathematics
Matrices
2022
JEE Advanced 2022 (Paper 2)
JEE Advanced Mathematics Question (2022) — Solution
Options
- A. A
- B. B
- C. C
- D. D
Answer
A. A
Step-by-step solution
Given, A = β 0 1 2 1 - 2 3 1 - 2 So, det A = - 1         . . . 1 Now for A 7 - β - 1 A 6 - β A 5 to be singular A 7 - β - 1 A 6 - β A 5 = 0 ⇒ A 5 A 2 - ( β - 1 ) A - β = 0             . . . 2 ⇒ A 5 A + I A - β l = 0 ∴ A 5 A + I A - β = 0 As A ≠ 0 A + 1 or A - β = 0 ⇒ β + 1 0 1 2 2 - 2 3 1 - 1 = 0 ⇒ A + I = - 4 ,   here  A ≠ 0   &   A + I ≠ 0 Now solving A - β l = 0 0 1 2 1 - β - 2 3 1 - 2 - β = 0 ⇒ 2 - 3 1 - β = 0 ⇒ 2 - 3 + 3 β = 0 ⇒    β = 1 3 ∴    9 β = 3
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