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*Let n be a positive integer and let Hn be the space of all n x n matrices A = (aij) with entries in R satisfying ajj = ars whenever i+j=r+s( i, j, r, s = 1, 2 ,...,n) then the dimention of Hn, as a vector space over R is*
a) n
b) n² - n + 1
c) 2n + 1
d) 2n - 1?
Most Upvoted Answer
*Let n be a positive integer and let Hn be the space of all n x n matr...
Explanation:

Definition of Hn:
- Let Hn be the space of all n x n matrices A = (aij) with entries in R satisfying ajj = ars whenever i+j=r+s for i, j, r, s = 1, 2, ..., n.

Dimension of Hn:
- To find the dimension of Hn, we need to determine the number of independent parameters needed to specify a matrix A in Hn.
- Let's consider the main diagonal entries a11, a22, ..., ann. These are independent as they do not depend on any other entries.
- Next, consider the entries above the main diagonal aij where i < j.="" these="" are="" also="" independent="" as="" they="" do="" not="" depend="" on="" the="" entries="" below="" the="" main="" />
- Therefore, the total number of independent entries above the main diagonal is 1 + 2 + ... + (n-1) = n(n-1)/2.

Total Dimension:
- The total number of independent entries in an n x n matrix is n².
- Subtracting the redundant entries in the main diagonal, we get n² - n.
- Finally, taking into account the condition ajj = ars, we have n independent entries on the main diagonal.
- Therefore, the dimension of Hn is n² - n + n = n².

Correct Answer:
- The dimension of Hn, as a vector space over R, is n².
So, the correct option is:
a) n²
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*Let n be a positive integer and let Hn be the space of all n x n matrices A = (aij) with entries in R satisfying ajj = ars whenever i+j=r+s( i, j, r, s = 1, 2 ,...,n) then the dimention of Hn, as a vector space over R is* a) nb) n² - n + 1c) 2n + 1d) 2n - 1?
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*Let n be a positive integer and let Hn be the space of all n x n matrices A = (aij) with entries in R satisfying ajj = ars whenever i+j=r+s( i, j, r, s = 1, 2 ,...,n) then the dimention of Hn, as a vector space over R is* a) nb) n² - n + 1c) 2n + 1d) 2n - 1? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about *Let n be a positive integer and let Hn be the space of all n x n matrices A = (aij) with entries in R satisfying ajj = ars whenever i+j=r+s( i, j, r, s = 1, 2 ,...,n) then the dimention of Hn, as a vector space over R is* a) nb) n² - n + 1c) 2n + 1d) 2n - 1? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for *Let n be a positive integer and let Hn be the space of all n x n matrices A = (aij) with entries in R satisfying ajj = ars whenever i+j=r+s( i, j, r, s = 1, 2 ,...,n) then the dimention of Hn, as a vector space over R is* a) nb) n² - n + 1c) 2n + 1d) 2n - 1?.
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