Matrices & Determinants Question - None

Question

Let β be a real number. Consider the matrix

A = $[\begin{matrix} β & 0 & 1 \\ 2 & 1 & - 2 \\ 3 & 1 & - 2 \end{matrix}]$

If A⁷ - (β-1)A⁶ - βA⁵ is a singular matrix, then the value of 9β is ____.

Accepted Answer

Given A = $[\begin{matrix} β & 0 & 1 \\ 2 & 1 & - 2 \\ 3 & 1 & - 2 \end{matrix}]$ . det(A) = β(-2 + 2) - 0 + 1(2 - 3) = -1. The expression is A⁵(A² - (β-1)A - βI) = A⁵(A - βI)(A + I). For this to be singular, det(A⁵(A - βI)(A + I)) = 0, which means det(A)⁵ * det(A - βI) * det(A + I) = 0. Since det(A) = -1, det(A)⁵ = -1 ≠ 0. Thus, det(A - βI) = 0 or det(A + I) = 0. det(A - βI) = det( $[\begin{matrix} 0 & 0 & 1 \\ 2 & 1 - β & - 2 \\ 3 & 1 & - 2 - β \end{matrix}]$ ) = 1(2 - 3(1-β)) = 2 - 3 + 3β = 3β - 1 = 0 ⇒ β = 1/3. det(A + I) = det( $[\begin{matrix} β + 1 & 0 & 1 \\ 2 & 2 & - 2 \\ 3 & 1 & - 1 \end{matrix}]$ ) = (β+1)(-2+2) + 1(2-6) = -4 ≠ 0. So β = 1/3. Then 9β = 9(1/3) = 3.

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