edited by
8,325 views
23 votes
23 votes

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?

  1. $S1$ and $S2$ both are true
  2. $S1$ is true, $S2$ is false
  3. $S1$ is false, $S2$ is true
  4. $S1$ and $S2$ both are false
edited by

6 Answers

26 votes
26 votes
A singular matrix is a square matrix whose determinant is 0. Whenever you solve question related to determinant, think of triangular matrix, like upper triangular matrix, because its determinant is just product of diagonal entries, and so easy to visualize. So definitely, in a singular matrix, one of the entries in diagonal must be zero. Similarly, in non-singular, none of the entries should be zero.

Now take any matrices and just check .

So we see that both S1 and S2 are true. So option (A) is correct.
edited by
9 votes
9 votes
ans should be A.

for S1 :  singular matrices matrix[0,0,0,1] + matrix[1,0,0,0] = matrix[1,0,0,1], which is non singular

for S2 : non-singular matrices matrix[1,0,0,1] + matrix[0,1,1,0] = matrix[1,1,1,1], which is singular
5 votes
5 votes

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

3 votes
3 votes
I think there is a contradiction  Let me explain with example
 If A = [ 2 10
            1  5]
B = [ 3 6
        2  4]
 Sum [ 5 16
           3  9]
| sum| = 9×5_16×3=- 3 which is not equal to zero . So it is non singular
edited by
Answer:

Related questions

38 votes
38 votes
9 answers
1
Kathleen asked Sep 14, 2014
12,472 views
How many $4$-digit even numbers have all $4$ digits distinct?$2240$$2296$$2620$$4536$
37 votes
37 votes
6 answers
4
Kathleen asked Sep 14, 2014
17,886 views
Which of the following statements is false?An unambiguous grammar has same leftmost and rightmost derivationAn LL(1) parser is a top-down parserLALR is more powerful than...