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 _____ .

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