Let A be a square matrix of order 3 and let O be the zero column matrix. For the homogeneous system \(AX = O\), consider the following statements:
(A) The system is always consistent.
(B) If \(|A| \ne 0\), then \(X = O\) is its only solution.
(C) If \(|A| = 0\), then the system has no solution.
(D) If \(X_1\) is a solution, then \(3X_1\) is also a solution.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (D) only
The zero column X = O always satisfies AX = O, so a homogeneous system always has at least this trivial solution.
If \(|A| \ne 0\), then \(A^{-1}\) exists:
\(X = A^{-1}O = O\)
If \(|A| = 0\), the system has infinitely many solutions, non-zero ones as well.
If \(AX_1 = O\), then:
\(A(3X_1) = 3(AX_1) = O\)
So the system is always consistent, has only X = O when \(|A| \ne 0\), has infinitely many solutions when \(|A| = 0\), and any multiple of a solution is again a solution.
Checking the statements
(A) Always consistent — Correct. X = O always satisfies AX = O.
(B) Only X = O when \(|A| \ne 0\) — Correct. \(X = A^{-1}O = O\).
(C) No solution when \(|A| = 0\) — Incorrect. X = O is still a solution, and there are infinitely many.
(D) \(3X_1\) is also a solution — Correct. \(A(3X_1) = 3(AX_1) = O\).