GATE CSE 2001 | Question: 1.1

Question

Consider the following statements:

• S1: The sum of two singular $n \times n$ matrices may be non-singular
• S2: The sum of two $n \times n$ non-singular matrices may be singular

Which one of the following statements is correct?

• A. $S1$ and $S2$ both are true
• B. $S1$ is true, $S2$ is false
• C. $S1$ is false, $S2$ is true
• D. $S1$ and $S2$ both are false

Comments

I think that without taking any example we can get the answer.   Following is the explaination for the two statements.

statement 1 : let us consider any two matrices of n-by-n which are singular, where n >= 2 (beacuse for n = 1 we can only have 1 matrix which is singular). Now, the addition of these two matrices may be singular or nonsingular. Assume that all such possible pairs of matrices yield the sum as non-singular, then certainly statement 1 is true. Now, assume only some possible pairs of n-by-n matrices yield the sum as non-singular while others as singular, then also statement 1 is true, due to the keyword "may be" in the statement. Now, assume the last possibility that all pairs of such matrices yield as singular matrix , then also statement 1 is true again due to "may be" keyword. Hence, statement 1 is true for all cases.

statement 2 : we can argue similarly that statement 2 is also true in all cases.

Hence option A is right answer. Please correct me if I am wrong.

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http://www.cse.iitd.ernet.in/~mittal/gate/gate_math_2001.html

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Both statements are possible.

S1 :

$A= \begin{bmatrix} 2 & 0 \\ 0 & 0 \end{bmatrix}$  $B= \begin{bmatrix} 0 & 0 \\ 0 & 2 \end{bmatrix}$

$ (A+B) = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$

S2 :

$A= \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$  $B= \begin{bmatrix} -2 & 0 \\ 0 & -2 \end{bmatrix}$

$ (A+B) = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Answer 1

$\boldsymbol{\textbf{A. Is correct option }}$

Comments

@Prince Sindhiya

Yes, you are right.

Using counterexample is a good way to solve such type of problems.

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Question is not asking for "always" the statements has to be true,so no  need to find counter example ..it is mentioned as "may be" even if one example statisfies ,statement will be true !!

Answer 2

So $S_1$ and $S_2$ both are true,