Question

If A and B are two n times n non-singular matrices, then

• A. AB is non-singular
• B. AB is singular
• C. (AB)^-1 = A^-1 B^-1
• D. (AB)^-1 does not exist

Hint:

The product of two non-singular matrices is always non-singular, and the inverse of a product AB is (AB)^-1 = B^-1 A^-1 .

Answer 1

The Correct Option is A

Step 1: Understanding non-singular matrices. 
A non-singular matrix is a square matrix that has an inverse. The product of two non-singular matrices is always non-singular.

Step 2: Analyzing the product of two matrices. 
If both A and B are non-singular matrices, then their product AB is also non-singular. This is because the determinant of AB is the product of the determinants of A and B, and the determinant of a non-singular matrix is non-zero. 

Step 3: Exploring other options. 
(A) AB is non-singular: Correct. The product of two non-singular matrices is non-singular. 
(B) AB is singular: Incorrect. This is not true if both matrices are non-singular. 
(C) (AB)^-1 = A^-1 B^-1 : Incorrect. The correct inverse of AB is (AB)^-1 = B^-1 A^-1 , not A^-1 B^-1 . 
(D) (AB)^-1 does not exist: Incorrect. The inverse of AB exists, as long as A and B are non-singular. 
Step 4: Conclusion. 
Thus, the correct answer is (A) AB is non-singular. 

Final Answer: AB is non-singular.