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The number of A in Tp such that the trace of A is not divisible by pp but det(A) is divisible by p is?
[Note: The trace of a matrix is the sum of its diagonal entries.]
  • a)
    (p−1) (p2−p+1)
  • b)
    p3 − (p−1)2
  • c)
    (p−1)2
  • d)
    (p−1)(p2−2)
Correct answer is option 'C'. Can you explain this answer?
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The number of A in Tp such that the trace of A is not divisible by pp ...
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The number of A in Tp such that the trace of A is not divisible by pp ...
Explanation:

Counting the number of matrices A
- The number of matrices A in Tp is p^2, as the order of the matrices is p x p.

Conditions for counting
- For trace of A to not be divisible by p, the diagonal entries of A should not be divisible by p.
- For det(A) to be divisible by p, the sum of the products of the entries of any permutation of the rows is divisible by p.

Counting the number of non-zero entries on the diagonal
- There are p - 1 non-zero entries on the diagonal (as 0 cannot be on the diagonal).

Counting the number of ways to choose the non-zero entries
- For each non-zero entry on the diagonal, there are p - 1 choices (since 0 is not an option).

Counting the number of ways to choose the non-diagonal entries
- For each non-diagonal entry, there are p choices.

Calculating the total number of matrices meeting the conditions
- The total number of matrices meeting the conditions is (p - 1) * (p - 1) * p = (p - 1)^2 * p = (p - 1)^2p.
Therefore, the correct answer is option 'C' which is (p-1)^2.
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The number of A in Tp such that the trace of A is not divisible by pp but det(A) is divisible by p is?[Note: The trace of a matrix is the sum of its diagonal entries.]a)(p−1) (p2−p+1)b)p3− (p−1)2c)(p−1)2d)(p−1)(p2−2)Correct answer is option 'C'. Can you explain this answer?
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The number of A in Tp such that the trace of A is not divisible by pp but det(A) is divisible by p is?[Note: The trace of a matrix is the sum of its diagonal entries.]a)(p−1) (p2−p+1)b)p3− (p−1)2c)(p−1)2d)(p−1)(p2−2)Correct answer is option 'C'. Can you explain this answer? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The number of A in Tp such that the trace of A is not divisible by pp but det(A) is divisible by p is?[Note: The trace of a matrix is the sum of its diagonal entries.]a)(p−1) (p2−p+1)b)p3− (p−1)2c)(p−1)2d)(p−1)(p2−2)Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of A in Tp such that the trace of A is not divisible by pp but det(A) is divisible by p is?[Note: The trace of a matrix is the sum of its diagonal entries.]a)(p−1) (p2−p+1)b)p3− (p−1)2c)(p−1)2d)(p−1)(p2−2)Correct answer is option 'C'. Can you explain this answer?.
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