Consider the following statements:
Which one of the following statements is correct?
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.
det (A + B) is not equal to det (A) + det (B), it means that
if A and B both are singular then, the determinant of sum of A and B may not to be equal to 0.
S1 is correct
Similarly, if A and B are non singular matrix, then det(A+B) may not be Non Zero, it can be Zero means Singular.
S2 is also correct
$\boldsymbol{\textbf{A. Is correct option }}$
@Prince Sindhiya
Yes, you are right.
Using counterexample is a good way to solve such type of problems.