Question

Matrix $A=[a_{ij}]_{3\times3}$ where } \[ a_{ij}= \begin{cases} i+j, & i \ne j \\ i-j, & i=j \end{cases} \] Find matrix $A$.

• (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
• (A) - (III), (B) - (I), (C) - (IV), (D) - (II)
• (A) - (III), (B) - (I), (C) - (II), (D) - (IV)
• (A) - (II), (B) - (IV), (C) - (I), (D) - (III)

Hint:

Always verify the order of the matrix \( n \) before applying determinant power rules like \( |kA| = k^n|A| \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
First, construct matrix \( A \) using the given conditions, then calculate its determinant. Properties of determinants: \( |kA| = k^n|A| \) for order \( n \), \( |adj A| = |A|^{n-1} \), and \( |A(adj A)| = |A|^n \).

Step 2: Detailed Explanation:
Constructing matrix \( A \): \[ A = \begin{bmatrix} 1+1 & 1+2 & 1+3 \\ 2-1 & 2+2 & 2+3 \\ 3-1 & 3-2 & 3+3 \end{bmatrix} = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 2 & 1 & 6 \end{bmatrix} \] Determinant \( |A| = 2(24-5) - 3(6-10) + 4(1-8) = 2(19) - 3(-4) + 4(-7) = 38 + 12 - 28 = 22 \).

• (A) \( |A| \): \( 22 \). Matches (III).
• (B) \( |2A| \): \( 2^3 |A| = 8 \times 22 = 176 \). Matches (I).
• (C) \( |adj A| \): \( |A|^{3-1} = 22^2 = 484 \). Matches (IV).
• (D) \( |A(adj A)| \): \( |A|^3 = 22^3 = 10648 \). Matches (II).

Step 3: Final Answer:
The matching is (A)-(III), (B)-(I), (C)-(IV), (D)-(II), which corresponds to option (B).