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Question:

Let D be the set of all 3x3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0. The number of matrices A in D for which the system of linear equations AXYZ = 100 has a unique solution, is

less than 4

at least 10

at least 4 but less than 7

atleast 7 but less than 10

Solution:

⎡⎢⎣0 a b⎤⎢⎣a 0 c⎦⎥⎦either b=0 or c=0 ⇒|A|≠0 ⇒2 matrices
⎡⎢⎣0 a b⎤⎢⎣a 1 c⎦⎥⎦either a=0 or c=0 ⇒|A|≠0 ⇒2 matrices
⎡⎢⎣1 a b⎤⎢⎣a 0 c⎦⎥⎦either a=0 or b=0 ⇒|A|≠0 ⇒2 matrices.
⎡⎢⎣1 a b⎤⎢⎣a 1 c⎦⎥⎦If a=b=0 ⇒|A|=0
If a=c=0 ⇒|A|=0
If b=c=0 ⇒|A|=0 ⇒there will be only 6 matrices.