Let A and B be square matrices of order 3. Consider the following statements:
(A) \(|A + B| = |A| + |B|\) for all such A and B
(B) If A is skew-symmetric, then \(|A| = 0\)
(C) If two columns of A are identical, then \(|A| = 0\)
(D) \(|3A| = 3|A|\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (C) only
For a skew-symmetric matrix, \(A^{T} = -A\). For order 3:
\(|A| = |A^{T}| = |-A| = (-1)^{3}|A| = -|A|\)
\(2|A| = 0\), so \(|A| = 0\)
Interchanging two identical columns leaves A unchanged but changes the sign of its determinant, so \(|A| = -|A|\) and \(|A| = 0\).
For a scalar multiple of order 3:
\(|3A| = 3^{3}|A| = 27|A|\)
For a sum, take A = B = I:
\(|A + B| = |2I| = 8\), while \(|A| + |B| = 2\)
So a skew-symmetric matrix of order 3 and a matrix with two identical columns both have determinant 0, while \(|3A| = 27|A|\) and \(|A + B|\) is in general not \(|A| + |B|\).
Checking the statements
(A) \(|A + B| = |A| + |B|\) — Incorrect. For A = B = I it gives 8 on the left and 2 on the right.
(B) Skew-symmetric gives \(|A| = 0\) — Correct. \(|A| = (-1)^{3}|A|\) forces \(|A| = 0\).
(C) Identical columns give \(|A| = 0\) — Correct. Swapping them gives \(|A| = -|A|\).
(D) \(|3A| = 3|A|\) — Incorrect. \(|3A| = 27|A|\) for order 3.