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The number of different n x 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 x n symmetric matrices with each element bei...
In a symmetric matrix, the lower triangle must be the mirror image of upper triangle using the diagonal as mirror. Diagonal elements may be anything. Therefore, when we are counting symmetric matrices we count how many ways are there to fill the upper triangle and diagonal elements. Since the first row has n elements, second (n - 1) elements, third row ( n - 2) elements and so on upto last row, one element.
Total number of elements in diagonal + upper triangle
Now, each one of these elements can be either 0 or 1. So total number of ways we can fill these elements is
Since there is no choice for lower triangleelements the answer is power (2, {n2 + n)/2).
Total number of elements in diagonal + upper triangle
Now, each one of these elements can be either 0 or 1. So total number of ways we can fill these elements is
Since there is no choice for lower triangleelements the answer is power (2, {n2 + n)/2).
Most Upvoted Answer
The number of different n x n symmetric matrices with each element bei...
Number of different n x n symmetric matrices with each element being either 0 or 1
To find the number of different n x n symmetric matrices with each element being either 0 or 1, we need to consider the properties of a symmetric matrix and the possible values for each element.
Properties of a symmetric matrix:
1. The diagonal elements of a symmetric matrix are always the same.
2. The elements above the diagonal are the mirror image of the elements below the diagonal.
Possible values for each element:
In this case, each element can take one of two values: 0 or 1.
Approach:
To count the number of different symmetric matrices, we need to consider the number of choices for each element.
Counting the diagonal elements:
Since the diagonal elements are always the same, there are only two choices for each diagonal element - 0 or 1. Therefore, there are 2^n choices for the diagonal elements.
Counting the elements above the diagonal:
Since the elements above the diagonal are the mirror image of the elements below the diagonal, we only need to count the choices for the elements below the diagonal. The elements above the diagonal will automatically be determined.
Counting the elements below the diagonal:
For each element below the diagonal, there are two choices - 0 or 1. However, we need to be careful not to count the diagonal elements again. Since there are n diagonal elements, we need to subtract n from the total number of elements below the diagonal. Therefore, there are 2^(n(n-1)/2) choices for the elements below the diagonal.
Total number of choices:
To find the total number of choices, we multiply the number of choices for the diagonal elements with the number of choices for the elements below the diagonal. Therefore, the total number of choices is 2^n * 2^(n(n-1)/2) = 2^(n + n(n-1)/2) = 2^(n^2 + n)/2.
Answer:
The number of different n x n symmetric matrices with each element being either 0 or 1 is power(2, (n^2 + n)/2), which is option 'C'.
To find the number of different n x n symmetric matrices with each element being either 0 or 1, we need to consider the properties of a symmetric matrix and the possible values for each element.
Properties of a symmetric matrix:
1. The diagonal elements of a symmetric matrix are always the same.
2. The elements above the diagonal are the mirror image of the elements below the diagonal.
Possible values for each element:
In this case, each element can take one of two values: 0 or 1.
Approach:
To count the number of different symmetric matrices, we need to consider the number of choices for each element.
Counting the diagonal elements:
Since the diagonal elements are always the same, there are only two choices for each diagonal element - 0 or 1. Therefore, there are 2^n choices for the diagonal elements.
Counting the elements above the diagonal:
Since the elements above the diagonal are the mirror image of the elements below the diagonal, we only need to count the choices for the elements below the diagonal. The elements above the diagonal will automatically be determined.
Counting the elements below the diagonal:
For each element below the diagonal, there are two choices - 0 or 1. However, we need to be careful not to count the diagonal elements again. Since there are n diagonal elements, we need to subtract n from the total number of elements below the diagonal. Therefore, there are 2^(n(n-1)/2) choices for the elements below the diagonal.
Total number of choices:
To find the total number of choices, we multiply the number of choices for the diagonal elements with the number of choices for the elements below the diagonal. Therefore, the total number of choices is 2^n * 2^(n(n-1)/2) = 2^(n + n(n-1)/2) = 2^(n^2 + n)/2.
Answer:
The number of different n x n symmetric matrices with each element being either 0 or 1 is power(2, (n^2 + n)/2), which is option 'C'.
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