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A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the determinant chosen is non-zero is
- a)3/16
- b)3/8
- c)1/4
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
A determinant is chosen at random from the set of all determinants of ...
To find the probability that the randomly chosen determinant is non-zero, we need to determine the total number of possible determinants and the number of non-zero determinants.
Total Number of Possible Determinants:
The determinant of a 2x2 matrix can be calculated using the formula:
det(A) = (a*d) - (b*c)
where A is the matrix and a, b, c, and d are its elements.
In this case, the elements can only be 0 or 1. So, each element has 2 possible choices. Therefore, the total number of possible determinants is 2^4 = 16.
Number of Non-zero Determinants:
To find the number of non-zero determinants, we need to consider the possible cases where the determinant is non-zero.
Case 1: Both diagonal elements are non-zero.
In this case, the determinant will be non-zero if either (a*d) or (b*c) is non-zero. There are 3 possibilities for each of these diagonal elements: (1,1), (1,0), and (0,1). Therefore, there are 3*3 = 9 possible determinants in this case.
Case 2: One diagonal element is zero and the other is non-zero.
In this case, the determinant will always be zero regardless of the other elements. Therefore, there are no possible determinants in this case.
Case 3: Both diagonal elements are zero.
In this case, the determinant will be zero regardless of the other elements. Therefore, there are no possible determinants in this case.
Therefore, the number of non-zero determinants is 9.
Probability of Non-zero Determinant:
The probability is calculated by dividing the number of non-zero determinants by the total number of possible determinants.
P(non-zero) = 9/16 = 3/8
Therefore, the correct answer is option 'B' - 3/8.
Total Number of Possible Determinants:
The determinant of a 2x2 matrix can be calculated using the formula:
det(A) = (a*d) - (b*c)
where A is the matrix and a, b, c, and d are its elements.
In this case, the elements can only be 0 or 1. So, each element has 2 possible choices. Therefore, the total number of possible determinants is 2^4 = 16.
Number of Non-zero Determinants:
To find the number of non-zero determinants, we need to consider the possible cases where the determinant is non-zero.
Case 1: Both diagonal elements are non-zero.
In this case, the determinant will be non-zero if either (a*d) or (b*c) is non-zero. There are 3 possibilities for each of these diagonal elements: (1,1), (1,0), and (0,1). Therefore, there are 3*3 = 9 possible determinants in this case.
Case 2: One diagonal element is zero and the other is non-zero.
In this case, the determinant will always be zero regardless of the other elements. Therefore, there are no possible determinants in this case.
Case 3: Both diagonal elements are zero.
In this case, the determinant will be zero regardless of the other elements. Therefore, there are no possible determinants in this case.
Therefore, the number of non-zero determinants is 9.
Probability of Non-zero Determinant:
The probability is calculated by dividing the number of non-zero determinants by the total number of possible determinants.
P(non-zero) = 9/16 = 3/8
Therefore, the correct answer is option 'B' - 3/8.
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Community Answer
A determinant is chosen at random from the set of all determinants of ...
Is it option c ? cause I get the option c but the answer shows it is b.
I doubt . cause my method looks right
I doubt . cause my method looks right
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A determinant is chosen at random from the set of all determinants of order 2 with elements 0 or 1 only. The probability that the determinant chosen is non-zero isa) 3/16 b) 3/8 c) 1/4 d) none of these Correct answer is option 'B'. Can you explain this answer?
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