Question

Let A and B are n x n matrices from which A is invertible. Suppose AB is singular. What conclusion can be made about the invertibility of B?

Answer 1

Answer: Matrix B is non- invertible.

Explanation:

A matrix is said to be be singular is its determinant is zero,

We know that if a matrix is singular then it is not invertible. (1)

Or if a matrix is invertible then it should be non-singular matrix. (2)

Given : A and B are n x n matrices from which A is invertible.

Then A must be non-singular matrix. ( from 2 )

If AB is singular.

Then either A is singular or B is singular but A is a non-singular matrix.

Then , matrix B should be a singular matrix. ( from 2 )

So Matrix B is non- invertible. ( from 1 )