If A and B are two non-singular matrices of order n, then which of the following statement/statements is/are not correct?
(A) AB is non-singular.
(B) AB is singular.
(C) \((AB)^{-1} = A^{-1}B^{-1}\)
(D) \((AB)^{-1}\) does not exist.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B), (C) and (D) only
Since A and B are non-singular, \(|A| \ne 0\) and \(|B| \ne 0\). So
\( \displaystyle |AB| = |A|\,|B| \ne 0 \)
By the reversal law,
\( \displaystyle (AB)^{-1} = B^{-1}A^{-1} \)
So AB is non-singular, \((AB)^{-1}\) exists and \((AB)^{-1} = B^{-1}A^{-1}\).
Checking the statements
(A) AB is non-singular. — Correct. \(|AB| = |A|\,|B| \ne 0\).
(B) AB is singular. — Incorrect. \(|AB| \ne 0\), so AB is not singular.
(C) \((AB)^{-1} = A^{-1}B^{-1}\) — Incorrect. The order is reversed: \((AB)^{-1} = B^{-1}A^{-1}\), which equals \(A^{-1}B^{-1}\) only when A and B commute.
(D) \((AB)^{-1}\) does not exist. — Incorrect. AB is non-singular, so its inverse exists.
The statements that are not correct are (B), (C) and (D).