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The number of different n × n symmetric matrices with each element being either 0 or 1 is: (Note: power(2, x) is same as 2x)
- a)power(2, n)
- b)power(2, n2)
- c)power(2, (n2 + n)/2)
- d)power(2, (n2 - n)/2)
Correct answer is option 'C'. Can you explain this answer?
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The number of different n × n symmetric matrices with each elemen...
We are considering a symmetric matrix (given in question). So, we need to look at half of elements ie. either upper or lower traingle i.e. A[i][j] = A[j][i] Hence, No. of elements = (n^2 + n)/2 Since, we have only two elements : 0 & 1 No. of choices = 2 Therefore, No. of possibilities = 2 ^ (No. of elements) = 2 ^ ((n^2 + n)/2) = power (2 , (n^2 + n)/2)
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The number of different n × n symmetric matrices with each elemen...
Explanation:
Total number of elements in a symmetric matrix:
- In a symmetric matrix, the number of elements above the main diagonal is the same as the number of elements below the main diagonal.
- So, the total number of unique elements in a symmetric nxn matrix is (n^2 + n)/2.
Each element can be either 0 or 1:
- Each element in the matrix can take one of two values - 0 or 1.
Number of different matrices:
- Since each element can take one of two values, the total number of different symmetric matrices can be calculated as 2 raised to the power of the total number of unique elements in the matrix.
- Therefore, the number of different n x n symmetric matrices with each element being either 0 or 1 is 2^((n^2 + n)/2) or power(2, (n^2 + n)/2).
Therefore, the correct answer is option C) power(2, (n^2 + n)/2).
Total number of elements in a symmetric matrix:
- In a symmetric matrix, the number of elements above the main diagonal is the same as the number of elements below the main diagonal.
- So, the total number of unique elements in a symmetric nxn matrix is (n^2 + n)/2.
Each element can be either 0 or 1:
- Each element in the matrix can take one of two values - 0 or 1.
Number of different matrices:
- Since each element can take one of two values, the total number of different symmetric matrices can be calculated as 2 raised to the power of the total number of unique elements in the matrix.
- Therefore, the number of different n x n symmetric matrices with each element being either 0 or 1 is 2^((n^2 + n)/2) or power(2, (n^2 + n)/2).
Therefore, the correct answer is option C) power(2, (n^2 + n)/2).
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