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:

A and B are non-singular matrices of order n.

This means $\det(A) \neq 0$ and $\det(B) \neq 0$, so $A^{-1}$ and $B^{-1}$ both exist.

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For statement (A): AB is non-singular.

Using the determinant property:

$\det(AB) = \det(A) \times \det(B)$

Since $\det(A) \neq 0$ and $\det(B) \neq 0$:

$\det(AB) \neq 0$

Therefore, AB is non-singular.

Statement (A) is correct.

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For statement (B): AB is singular.

From above, AB is non-singular.

Statement (B) is not correct.

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For statement (C): $(AB)^{-1} = A^{-1}B^{-1}$

The inverse of a product reverses the order:

$(AB)^{-1} = B^{-1}A^{-1}$

Statement (C) is not correct.

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For statement (D): $(AB)^{-1}$ does not exist.

Since AB is non-singular, $(AB)^{-1}$ exists.

Statement (D) is not correct.

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The statements that are not correct are (B), (C), and (D).