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Eigenvalues Class 12th

Let P = [[-30, 20, 56],,] and A = [[2, 7, ω²], [-1, -ω, 1], [0, -ω, -ω+1]] where ω = (-1 + i√3)/2, and I be the identity matrix of order 3. If the determinant of the matrix (P?¹AP - I)² is αω², then the value of α is equal to.
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R
Raj Pandey

Contributor-Level 9

(P? ¹AP - I)²
= (P? ¹AP - I) (P? ¹AP - I)
= P? ¹A (PP? ¹)AP - P? ¹AP - P? ¹AP + I
= P? ¹A²P - 2P? ¹AP + I
= P? ¹ (A² - 2A + I)P = P? ¹ (A - I)²P
| (P? ¹AP - I)²| = |P? ¹ (A - I)²P| = |P? ¹| | (A - I)²| |P| = | (A - I)²| = |A - I|²
A - I = [1, 7, w²], [-1, w², 1], [0, -w, -w]
|A - I| = 1 (-w³ + w) - 7 (w) + w² (w) = -w³ + w - 7w + w³ = -6w.
|A - I|² = (-6w)² = 36w².

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