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Let P be an n x n idempotent matrix, that is, P2 = P. Which of the following is FALSE?
- a)AT is idempotent
- b)The possible eigenvalues of P are 0 or 1.
- c)The non diagonal entries of P can be zero.
- d)There are infinite number of n x n non singular matrices that are idempotent.
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Let P be an n x n idempotent matrix, that is, P2 = P. Which of the fol...
If P is non singular, there exist P-1
P2 = P
p -lp 2 = p-l p
P = I
So, P is identity matrix.
Alternate: As P2 = P
PP = P
(pp)T = pT
pTpT = pT
( pt)2 = pt
PT is idempotent
option (a) is discard
P2 = P
as matrix P satisfies characteristic polynomial, by Caylay Hamilton theorem
option (b) discard
take
Then P2 = P
option (c) discard.
P2 = P
p -lp 2 = p-l p
P = I
So, P is identity matrix.
Alternate: As P2 = P
PP = P
(pp)T = pT
pTpT = pT
( pt)2 = pt
PT is idempotent
option (a) is discard
P2 = P
as matrix P satisfies characteristic polynomial, by Caylay Hamilton theorem
option (b) discard
take
Then P2 = Poption (c) discard.
Most Upvoted Answer
Let P be an n x n idempotent matrix, that is, P2 = P. Which of the fol...
Statement: There are an infinite number of n x n non-singular matrices that are idempotent.
Explanation:
An idempotent matrix is a square matrix that remains unchanged when multiplied by itself. In other words, if P is an idempotent matrix, then P^2 = P.
To prove that there are an infinite number of n x n non-singular matrices that are idempotent, we need to show that there are infinitely many distinct idempotent matrices.
Proof:
Let's consider a matrix A = [a_ij] such that:
a_ij = 1 if i = j
a_ij = 0 if i ≠ j
This is a diagonal matrix with all diagonal entries as 1 and all non-diagonal entries as 0.
Now, let's consider the matrix B = 2A - I, where I is the identity matrix.
Claim 1: B is idempotent.
Proof:
B^2 = (2A - I)^2
= (2A - I)(2A - I)
= 4A^2 - 2A - 2A + I
= 4A - 4A + I
= I
Therefore, B^2 = B, which implies that B is idempotent.
Claim 2: B is non-singular.
Proof:
To show that B is non-singular, we need to prove that the determinant of B is non-zero.
det(B) = det(2A - I)
= 2^n * det(A) - det(I)
= 2^n - 1
Since n is a positive integer, 2^n is always greater than 1. Therefore, det(B) = 2^n - 1 ≠ 0.
Hence, B is non-singular.
Conclusion:
We have shown that the matrix B = 2A - I is both idempotent and non-singular for any positive integer n. Since there are infinitely many positive integers, there are an infinite number of n x n non-singular matrices that are idempotent.
Therefore, the statement "There are an infinite number of n x n non-singular matrices that are idempotent" is TRUE, and option D is FALSE.
Explanation:
An idempotent matrix is a square matrix that remains unchanged when multiplied by itself. In other words, if P is an idempotent matrix, then P^2 = P.
To prove that there are an infinite number of n x n non-singular matrices that are idempotent, we need to show that there are infinitely many distinct idempotent matrices.
Proof:
Let's consider a matrix A = [a_ij] such that:
a_ij = 1 if i = j
a_ij = 0 if i ≠ j
This is a diagonal matrix with all diagonal entries as 1 and all non-diagonal entries as 0.
Now, let's consider the matrix B = 2A - I, where I is the identity matrix.
Claim 1: B is idempotent.
Proof:
B^2 = (2A - I)^2
= (2A - I)(2A - I)
= 4A^2 - 2A - 2A + I
= 4A - 4A + I
= I
Therefore, B^2 = B, which implies that B is idempotent.
Claim 2: B is non-singular.
Proof:
To show that B is non-singular, we need to prove that the determinant of B is non-zero.
det(B) = det(2A - I)
= 2^n * det(A) - det(I)
= 2^n - 1
Since n is a positive integer, 2^n is always greater than 1. Therefore, det(B) = 2^n - 1 ≠ 0.
Hence, B is non-singular.
Conclusion:
We have shown that the matrix B = 2A - I is both idempotent and non-singular for any positive integer n. Since there are infinitely many positive integers, there are an infinite number of n x n non-singular matrices that are idempotent.
Therefore, the statement "There are an infinite number of n x n non-singular matrices that are idempotent" is TRUE, and option D is FALSE.
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Let P be an n x n idempotent matrix, that is, P2 = P. Which of the following is FALSE?a)AT is idempotentb)The possible eigenvalues of P are 0 or 1.c)The non diagonal entries of P can be zero.d)There are infinite number of n x n non singular matrices that are idempotent.Correct answer is option 'D'. Can you explain this answer?
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Let P be an n x n idempotent matrix, that is, P2 = P. Which of the following is FALSE?a)AT is idempotentb)The possible eigenvalues of P are 0 or 1.c)The non diagonal entries of P can be zero.d)There are infinite number of n x n non singular matrices that are idempotent.Correct answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let P be an n x n idempotent matrix, that is, P2 = P. Which of the following is FALSE?a)AT is idempotentb)The possible eigenvalues of P are 0 or 1.c)The non diagonal entries of P can be zero.d)There are infinite number of n x n non singular matrices that are idempotent.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let P be an n x n idempotent matrix, that is, P2 = P. Which of the following is FALSE?a)AT is idempotentb)The possible eigenvalues of P are 0 or 1.c)The non diagonal entries of P can be zero.d)There are infinite number of n x n non singular matrices that are idempotent.Correct answer is option 'D'. Can you explain this answer?.
Let P be an n x n idempotent matrix, that is, P2 = P. Which of the following is FALSE?a)AT is idempotentb)The possible eigenvalues of P are 0 or 1.c)The non diagonal entries of P can be zero.d)There are infinite number of n x n non singular matrices that are idempotent.Correct answer is option 'D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let P be an n x n idempotent matrix, that is, P2 = P. Which of the following is FALSE?a)AT is idempotentb)The possible eigenvalues of P are 0 or 1.c)The non diagonal entries of P can be zero.d)There are infinite number of n x n non singular matrices that are idempotent.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let P be an n x n idempotent matrix, that is, P2 = P. Which of the following is FALSE?a)AT is idempotentb)The possible eigenvalues of P are 0 or 1.c)The non diagonal entries of P can be zero.d)There are infinite number of n x n non singular matrices that are idempotent.Correct answer is option 'D'. Can you explain this answer?.
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