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Matrices & Determinants: Pmatrix Pmatrix Integers Identity Matrix Equals

JEE Maths question with a full step-by-step solution.

Question
If M=(112021102)M = \begin{pmatrix} 1 & 1 & 2 \\ 0 & 2 & 1 \\ 1 & 0 & 2 \end{pmatrix} and M3=(αM−I)(βM−I)M^3 = (\alpha M - I)(\beta M - I), where α,β\alpha, \beta are integers and II is the 3×33\times 3 identity matrix, then (α+β)(\alpha + \beta) equals:
A44
B55
C66correct
D77
Solution
Step 1: Form the characteristic equation
∣M−λI∣=0  ⟹  λ3−5λ2+6λ−1=0|M - \lambda I| = 0 \implies \lambda^3 - 5\lambda^2 + 6\lambda - 1 = 0
 Step 2: Apply the Cayley-Hamilton theorem
M3−5M2+6M−I=0  ⟹  M3=5M2−6M+IM^3 - 5M^2 + 6M - I = 0 \implies M^3 = 5M^2 - 6M + I
 Step 3: Expand the proposed factorization
(αM−I)(βM−I)=αβM2−(α+β)M+I(\alpha M - I)(\beta M - I) = \alpha\beta M^2 - (\alpha + \beta)M + I
 Comparing with M3=5M2−6M+IM^3 = 5M^2 - 6M + I gives αβ=5\alpha\beta = 5 and α+β=6\alpha + \beta = 6. Answer: (3)
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