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+11 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
asked in Linear Algebra by Veteran (69k points)
edited by | 894 views

4 Answers

+14 votes
Best answer
Yes A is correct option!, Both statements are True


$A=\begin{matrix} 2&10 \\ 1&5 \end{matrix}$$\qquad B=\begin{matrix} 3&6 \\ 2&4 \end{matrix}$


$|A|=0 \quad |B|=0$




$A=\begin{matrix} 1&0 \\ 0&1 \end{matrix}$$\qquad B=\begin{matrix} 0&1 \\ 1&0 \end{matrix}$


$|A|=1 \quad |B|={-1}$

answered by Veteran (18.3k points)
edited by
+6 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
answered by Boss (8.5k points)
but matrices should be square ..?
yes, that is true
+2 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
answered by (41 points)
edited by
+2 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.
answered by Active (2.3k points)
edited by

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