JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{pmatrix}$ . Then the sum of the diagonal elements of the matrix $(A + I)^{11}$ is equal to:

Choose the correct answer:

Explanation

Solution

Matrix $(A+I)^{11}$ ke diagonal elements ka sum nikalne ka matlab hai uski Trace nikalna.

Step 1: Matrix $A$ ki Eigenvalues nikalna

Characteristic equation $|A - \lambda I| = 0$ ka use karke:

\begin{vmatrix} 1-\lambda & 0 & 0 \\ 0 & 4-\lambda & -1 \\ 0 & 12 & -3-\lambda \end{vmatrix} = 0

(1-\lambda) [(4-\lambda)(-3-\lambda) - (-12)] = 0

(1-\lambda) [\lambda^2 - \lambda - 12 + 12] = 0

(1-\lambda) (\lambda^2 - \lambda) = 0

(1-\lambda) \lambda (\lambda - 1) = 0

Eigenvalues ( $\lambda$ ) hain: $1, 1, 0$ .

Step 2: Matrix $(A+I)$ ki Eigenvalues

Yadi $A$ ki eigenvalues $\lambda$ hain, toh $(A+I)$ ki eigenvalues $(\lambda + 1)$ hongi:

\text{Eigenvalues of } (A+I) = (1+1), (1+1), (0+1) = 2, 2, 1

Step 3: Matrix $(A+I)^{11}$ ki Eigenvalues

(A+I)^{11} \text{ ki eigenvalues} = 2^{11}, 2^{11}, 1^{11}

= 2048, 2048, 1

Step 4: Sum of diagonal elements (Trace)

Matrix ki trace uski eigenvalues ka sum hoti hai:

\text{Sum} = 2048 + 2048 + 1

\text{Sum} = 4096 + 1 = 4097

Correct Option: (3) 4097