Let A be a square matrix of order 3 such that \(|A| = 6\).
Match List-I with List-II
| List-I | List-II | ||
| (A) | The determinant of the matrix obtained from A by interchanging two of its rows | (I) | 162 |
| (B) | The determinant of the matrix obtained from A by multiplying each element of one row by 3 | (II) | 6 |
| (C) | \(|3A|\) | (III) | \(-6\) |
| (D) | The determinant of the matrix obtained from A by applying \(R_1 \to R_1 + 2R_2\) | (IV) | 18 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
Each value follows from an elementary property of determinants.
(A) The determinant of the matrix obtained from A by interchanging two of its rows
This matches (III) −6. Interchanging two rows changes only the sign:
\(-|A| = -6\)
(B) The determinant of the matrix obtained from A by multiplying each element of one row by 3
This matches (IV) 18. Multiplying one row by 3 multiplies the determinant by 3 once:
\(3|A| = 3 \times 6 = 18\)
(C) \(|3A|\)
This matches (I) 162. In 3A every one of the 3 rows is multiplied by 3:
\(|3A| = 3^3|A| = 27 \times 6 = 162\)
(D) The determinant of the matrix obtained from A by applying \(R_1 \to R_1 + 2R_2\)
This matches (II) 6. Adding a multiple of one row to another row leaves the determinant unchanged:
\(|A| = 6\)