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The dimension of a vector space of 10x10 real matrices with row sum, column sum, trace and sum of all entire as zero is A. 79 B. 78 C. 80 D. None?
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The dimension of a vector space of 10x10 real matrices with row sum, c...
Dimension of the Vector Space of 10x10 Real Matrices
There are certain conditions given for the matrices in the vector space:
1. Row sum, column sum, trace, and sum of all entries are zero.
Calculating the Dimension
To calculate the dimension of the vector space of 10x10 real matrices that satisfy the given conditions, we need to consider the degrees of freedom for each condition:
1. Row sum: There are 9 independent row sums (since the sum of all entries in a row must be zero).
2. Column sum: Similarly, there are 9 independent column sums.
3. Trace: The trace of a matrix is the sum of its diagonal elements, so there are 9 independent choices for the trace.
4. Sum of all entries: Given that the sum of all entries in the matrix is zero, there is only one constraint on the sum of all entries.
Calculating the Total Degree of Freedom
Adding up the degrees of freedom from each condition, we get:
9 (row sums) + 9 (column sums) + 9 (trace) - 1 (sum of all entries) = 26
Therefore, the dimension of the vector space of 10x10 real matrices satisfying the given conditions is 26.
Therefore, the correct answer is None as none of the options provided match the calculated dimension of the vector space.
There are certain conditions given for the matrices in the vector space:
1. Row sum, column sum, trace, and sum of all entries are zero.
Calculating the Dimension
To calculate the dimension of the vector space of 10x10 real matrices that satisfy the given conditions, we need to consider the degrees of freedom for each condition:
1. Row sum: There are 9 independent row sums (since the sum of all entries in a row must be zero).
2. Column sum: Similarly, there are 9 independent column sums.
3. Trace: The trace of a matrix is the sum of its diagonal elements, so there are 9 independent choices for the trace.
4. Sum of all entries: Given that the sum of all entries in the matrix is zero, there is only one constraint on the sum of all entries.
Calculating the Total Degree of Freedom
Adding up the degrees of freedom from each condition, we get:
9 (row sums) + 9 (column sums) + 9 (trace) - 1 (sum of all entries) = 26
Therefore, the dimension of the vector space of 10x10 real matrices satisfying the given conditions is 26.
Therefore, the correct answer is None as none of the options provided match the calculated dimension of the vector space.
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The dimension of a vector space of 10x10 real matrices with row sum, column sum, trace and sum of all entire as zero is A. 79 B. 78 C. 80 D. None?
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The dimension of a vector space of 10x10 real matrices with row sum, column sum, trace and sum of all entire as zero is A. 79 B. 78 C. 80 D. None? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about The dimension of a vector space of 10x10 real matrices with row sum, column sum, trace and sum of all entire as zero is A. 79 B. 78 C. 80 D. None? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The dimension of a vector space of 10x10 real matrices with row sum, column sum, trace and sum of all entire as zero is A. 79 B. 78 C. 80 D. None?.
The dimension of a vector space of 10x10 real matrices with row sum, column sum, trace and sum of all entire as zero is A. 79 B. 78 C. 80 D. None? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about The dimension of a vector space of 10x10 real matrices with row sum, column sum, trace and sum of all entire as zero is A. 79 B. 78 C. 80 D. None? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The dimension of a vector space of 10x10 real matrices with row sum, column sum, trace and sum of all entire as zero is A. 79 B. 78 C. 80 D. None?.
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