Let A be a square matrix of order 3 such that \(|A| = -2\). Which of the following statements are correct?
(A) \(|A'| = -2\)
(B) \(|2A| = -4\)
(C) \(|-A| = 2\)
(D) \(|AA'| = 4\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (C) and (D) only
For square matrices of order 3, three results are used: the transpose has the same determinant, \(|kA| = k^3|A|\) because each of the 3 rows is multiplied by k, and the product rule \(|AB| = |A||B|\).
\(|A'| = |A| = -2\)
\(|2A| = 2^3 \times (-2) = -16\)
\(|-A| = (-1)^3 \times (-2) = 2\)
\(|AA'| = |A||A'| = (-2) \times (-2) = 4\)
So \(|A'| = -2\), \(|2A| = -16\), \(|-A| = 2\) and \(|AA'| = 4\).
Checking the statements
(A) \(|A'| = -2\) — Correct. Transposing does not change the determinant.
(B) \(|2A| = -4\) — Incorrect. \(|2A| = 2^3 \times (-2) = -16\).
(C) \(|-A| = 2\) — Correct. \((-1)^3 \times (-2) = 2\).
(D) \(|AA'| = 4\) — Correct. \(|A||A'| = (-2) \times (-2) = 4\).