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JEE Mains 31 Jan 2023 Shift 1 Paper

Section: Mathematics

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Question : 67 of 90

Marks: +1, -0

Let

A=\begin{array}{ccc}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{array}

. Then the sum of the diagonal elements of the matrix

(A+I)^{11}

[31-Jan-2023 Shift 1]

6144
4094
4097
2050

Solution:

\;A^2=\begin{bmatrix}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{bmatrix}\begin{bmatrix}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{bmatrix}

\;\; =\begin{bmatrix}1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3\end{bmatrix}= A

\;\Rightarrow A^3 = A^4 = \dots = A

\;(A+I)^{11}=^{11}C_0 A^{11}+^{11}C_1 A^{10}+\dots+^{11}C_{10} A+^{11}C_{11} I

\; = \left(^{11}C_0+^{11}C_1+\dots+^{11}C_{10}\right) A + I

\; = (2^{11}-1) A + I = 2047 A + I

\; \therefore \; \text{ Sum of diagonal elements } \; = 2047(1+4-3)+3

\; = 4094+3 = 4097

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