Let A be a singular square matrix of order 3. Consider the following statements:
(A) \(|\mathrm{adj}\,A| = 0\)
(B) \(A(\mathrm{adj}\,A) = O\), the zero matrix
(C) \(A^{-1}\) exists
(D) The system \(AX = B\) has a unique solution for every column matrix B
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
A singular matrix has \(|A| = 0\). For order 3:
\(|\mathrm{adj}\,A| = |A|^{2} = 0\)
\(A(\mathrm{adj}\,A) = |A|\,I = O\)
A matrix has an inverse only when its determinant is not zero, and AX = B has a unique solution only when \(|A| \ne 0\).
So \(|\mathrm{adj}\,A| = 0\), \(A(\mathrm{adj}\,A) = O\), \(A^{-1}\) does not exist, and AX = B never has a unique solution.
Checking the statements
(A) \(|\mathrm{adj}\,A| = 0\) — Correct. \(|\mathrm{adj}\,A| = |A|^{2} = 0\).
(B) \(A(\mathrm{adj}\,A) = O\) — Correct. \(A(\mathrm{adj}\,A) = |A|\,I = 0 \cdot I\).
(C) \(A^{-1}\) exists — Incorrect. \(A^{-1} = \frac{1}{|A|}\mathrm{adj}\,A\) needs \(|A| \ne 0\).
(D) Unique solution for every B — Incorrect. With \(|A| = 0\) there is either no solution or infinitely many.