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
Given: A and B are non-singular (invertible) n×n matrices.
(A) AB is non-singular . If $A$ and $B$ are non-singular, then $|A| \neq 0$ and $|B| \neq 0$. The determinant of their product is $|AB| = |A| \cdot |B|$. Since neither determinant is zero, $|AB| \neq 0$. Therefore, $AB$ is non-singular. This statement is correct.
(B) AB is singular . As established above, $AB$ is non-singular. Therefore, stating that $AB$ is singular is incorrect.
(C) $(AB)^{-1} = A^{-1} B^{-1}$: The property for the inverse of a product is $(AB)^{-1} = B^{-1}A^{-1}$ (the reversal law). The expression $A^{-1}B^{-1}$ is generally not equal to $(AB)^{-1}$ unless the matrices commute ($AB = BA$). Therefore, this statement is incorrect.
(D) $(AB)^{-1}$ does not exist . Since $AB$ is non-singular, its inverse must exist. Therefore, stating that it does not exist is incorrect.