Matrices & DeterminantsmediumFree

Matrices & Determinants — JEE Maths practice question

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

Question

M = \begin{pmatrix} 1 & 1 & 2 \\ 0 & 2 & 1 \\ 1 & 0 & 2 \end{pmatrix}

and

M^3 = (\alpha M - I)(\beta M - I)

, where

\alpha, \beta

are integers and

I

is the

3\times 3

identity matrix, then

(\alpha + \beta)

equals:

4

5

6

correct

7

Solution

Step 1: Form the characteristic equation

|M - \lambda I| = 0 \implies \lambda^3 - 5\lambda^2 + 6\lambda - 1 = 0

Step 2: Apply the Cayley-Hamilton theorem

M^3 - 5M^2 + 6M - I = 0 \implies M^3 = 5M^2 - 6M + I

Step 3: Expand the proposed factorization

(\alpha M - I)(\beta M - I) = \alpha\beta M^2 - (\alpha + \beta)M + I

Comparing with

M^3 = 5M^2 - 6M + I

gives

\alpha\beta = 5

and

\alpha + \beta = 6

. Answer: (3)

Still stuck on this question?Ask your doubt on WhatsApp

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.