1. Total number of determinants:
A matrix of order 2×2 has 4 elements:
Since each element (a,b,c,d) can be either 0 or 1, there are 2 choices for each position.
Total possible determinants =2×2×2×2=24=16.
2. Calculation of the determinant:
The value of the determinant is given by:
Since the elements are only 0 or 1, the possible values for ad are {0,1} and for bc are {0,1}.
The possible values for the determinant Δ are {1−0,0−1,0−0,1−1}, which simplifies to {1,−1,0}.
3. Finding non-zero determinants (Δ=0):
The determinant is non-zero if:
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Case 1: Δ=1
This happens when ad=1 and bc=0.
ad=1 only if a=1,d=1 (1 way).
bc=0 if (b,c) is (0,0),(0,1), or (1,0) (3 ways).
Total ways for Δ=1: 1×3=3.
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Case 2: Δ=−1
This happens when ad=0 and bc=1.
ad=0 if (a,d) is (0,0),(0,1), or (1,0) (3 ways).
bc=1 only if b=1,c=1 (1 way).
Total ways for Δ=−1: 3×1=3.
Total non-zero outcomes =3+3=6.
4. Probability Calculation:
P(Non-zero)=Total number of outcomesNumber of favorable outcomes
Conclusion:
The total number of such matrices is 16, and 6 of them have a non-zero determinant.
Correct Option: (b)
