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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 | 985 views

Yes A is correct option!, Both statements are True

$S_{1}:\text{True}$

$A=\begin{matrix} 2&10 \\ 1&5 \end{matrix}$$\qquad B=\begin{matrix} 3&6 \\ 2&4 \end{matrix} |A|=0 \quad |B|=0 |A+B|={-3} S_{2}:\text{True} A=\begin{matrix} 1&0 \\ 0&1 \end{matrix}$$\qquad B=\begin{matrix} 0&1 \\ 1&0 \end{matrix}$

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

$|A+B|=0$
edited
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
0
but matrices should be square ..?
0
yes, that is true
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
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

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

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

answered ago by Junior (725 points)