GATE 2001 CS question 1.1 (Matrix Algebra)

1.1 Consider the following statements:
S1: The sum of two singular n X n matrices may be non-singular.
S2: The sum of two n X n non-singular matrices may be singular.
Which of the following statements is correct?
(a) S1 and S2 are both true. (b) S1 is true, S2 is false.
(c) S1 is false, S2 is true. (d) S1 and S2 are both false.

This is one problem where getting too theoretical can get you stuck. Instead, let's try to come up with simple examples. The one I immediately thought of for S1 was:

1	1
1	1

and

1	-1
-1	1

Both these matrices are singular. What does 'singular' mean? It simply means that the determinant of the matrix is zero (if you don't remember how the determinant of a matrix is found, please see this post). Now, what is their matrix sum? It is:

2	0
0	2

Now quick, what is the determinant value of this one? It's 2 X 2 - 0 X 0 = 4. Clearly, this sum is not 0, so the resulting matrix is non-singular. So, we've proved that S1 is true. The second case is still easier. Let's take the matrices

1	0
0	1

and

-1	0
0	-1

Now, both of them have a determinant value of 1, and so they are not singular. What's their matrix sum then?

0	0
0	0

Wow! That's very clearly a singular matrix - it's determinant can't be anything other than 0. So, two non-singular matrices can also give a singular matrix on addition. This means that S2 is also true.

So, the answer is: (a) S1 and S2 are both true