Matrices & DeterminantseasyFree

Matrices & Determinants: Let Matrix Order Defined

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

Question

Let

A

be a matrix of order 3 defined by

A = [a_{ij}]_{3\times 3}

, where

a_{ij} = \displaystyle\lim_{x \to 0}\frac{\sin(ix)}{\tan(jx)}

for all

1 \le i, j \le 3

. Then

A^2

is:

4A

3A

correct

2A

A

Solution

Step 1: Identify the entries

\lim_{x \to 0}\frac{\sin(ix)}{\tan(jx)} = \frac{i}{j} \implies a_{ij} = \frac{i}{j}

Step 2: Compute a general entry of

A^2

(A^2)_{ik} = \sum_{j=1}^{3} a_{ij}\,a_{jk} = \sum_{j=1}^{3}\frac{i}{j}\cdot\frac{j}{k} = \sum_{j=1}^{3}\frac{i}{k} = 3\cdot\frac{i}{k} = 3\,a_{ik}

Therefore

A^2 = 3A

. Answer: (2)

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