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Question

# Let A be the set of all 3×3 symmetric matrices, all of whose entries are either 0 or 1, five of these are 1 and four of them are 0 Consider the system of linear equations A⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣100⎤⎥⎦, then which of the following is/are correct?

A
The total number of possible matrices in A is 12.
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B
The number of matrices A in set A for which the system of linear equations has a unique solution is 6.
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C
The number of matrices A in set A for which the system of linear equations has a unique solution is 4.
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D
The number of matrices A in set A for which the system of linear equations is inconsistent is more than 2.
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Solution

## The correct options are A The total number of possible matrices in A is 12. B The number of matrices A in set A for which the system of linear equations has a unique solution is 6. D The number of matrices A in set A for which the system of linear equations is inconsistent is more than 2. Let the matrix A=⎡⎢⎣abclmnpqr⎤⎥⎦ Given: Five entries as 1 and remaining four entries as 0. Since, the matrix is symmetric, it must have even number of zeros for i≠j. Therefore, there are two cases: Case (1) Two entries in diagonal are zero. We can select two places from three (in diagonal) in 3C2 ways Now, We have to select elements for upper triangle. For upper triangle, we have three places of which one entry is ′0′ and two are ′1′. Therefore, one place from three can be selected in 3C1 ways. Hence, the number of matrices = 3C2× 3C1=9. Case (2) If all the entries in the principal diagonal are 1, we have two ′0′ and one ′1′ in upper triangle. Hence, the number of matrices =3. Therefore, the total number of matrices in A is 3+9=12 Now, for the sytstem of equation to have unique solution |A|≠0 Consider the matrix A=⎡⎢⎣0aba0cbc1⎤⎥⎦ |A|=a2 So, |A|≠0 if either b=0 or c=0 ∴ number of matrices =2 Now, Consider the matrix A=⎡⎢⎣0aba1cbc0⎤⎥⎦ |A|=b2 So, |A|≠0 if either a=0 or c=0 ∴ number of matrices =2 Now, Consider the matrix A=⎡⎢⎣1aba0cbc0⎤⎥⎦ Clearly, |A|≠0 if either a=0 or b=0 ∴ number of matrices =2 Now, Consider the matrix A=⎡⎢⎣1aba1cbc1⎤⎥⎦ Clearly, a=b=0,c=1⇒|A|=0 a=c=0,b=1⇒|A|=0 b=c=0,a=1⇒|A|=0 ∴ number of matrices in this case =0. Hence, total number of matrices having unique solutions is 6. The six matrices A for which |A|=0 are: A=⎡⎢⎣001001111⎤⎥⎦⇒ inconsistent A=⎡⎢⎣010111010⎤⎥⎦⇒ inconsistent A=⎡⎢⎣111100100⎤⎥⎦⇒ infinite solutions A=⎡⎢⎣110110001⎤⎥⎦⇒ inconsistent A=⎡⎢⎣101010101⎤⎥⎦⇒ inconsistent A=⎡⎢⎣100011011⎤⎥⎦⇒ inconsistent

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