Let A be a square matrix of order 3 with \(|A| = 4\), and let B be the matrix obtained from A as described in List-I. Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| B is obtained from A by | \(|B|\) | ||
| (A) | Interchanging its first and third rows | (I) | 12 |
| (B) | Multiplying its second row by 3 | (II) | 4 |
| (C) | Multiplying every element by 2 | (III) | \(-4\) |
| (D) | Applying \(R_2 \to R_2 + 5R_1\) | (IV) | 32 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(I), (C)-(IV), (D)-(II)
Each item uses one property of determinants, with \(|A| = 4\).
(A) Interchanging its first and third rows
This matches (III) \(-4\).
Interchanging two rows changes the sign of the determinant:
\(|B| = -|A| = -4\)
(B) Multiplying its second row by 3
This matches (I) 12.
Multiplying one row by 3 multiplies the determinant by 3:
\(|B| = 3 \times 4 = 12\)
(C) Multiplying every element by 2
This matches (IV) 32.
Here B = 2A, and each of the 3 rows gives a factor 2:
\(|B| = 2^{3} \times 4 = 32\)
(D) Applying \(R_2 \to R_2 + 5R_1\)
This matches (II) 4.
Adding a multiple of one row to another row does not change the determinant:
\(|B| = |A| = 4\)