Mathematics Exam > Mathematics Questions > The sum of two idempotent matrices A and B is...
Start Learning for Free
The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?
Correct answer is '0'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
The sum of two idempotent matrices A and B is idempotent and if AB + B...
Given A2 = A and B2 = B
and (A + B) = ( A + B)2 = A2+ B2 + AB + BA
= A + B + AB + BA
⇒ A + B = A + B + AB + BA
⇒ AB + BA = 0 (0: Null matrix)
⇒ C = 0 ⇒ det C =0
Most Upvoted Answer
The sum of two idempotent matrices A and B is idempotent and if AB + B...
Idempotent Matrices
- An idempotent matrix is a square matrix that, when multiplied by itself, results in the same matrix.
- Mathematically, for a matrix A to be idempotent, it must satisfy the condition A^2 = A.
Sum of Idempotent Matrices
- Let A and B be two idempotent matrices.
- We need to show that the sum of A and B, denoted as C = A + B, is also idempotent.
- To prove this, we can calculate the square of C: C^2 = (A + B)^2.
Calculating (A + B)^2
- Expanding (A + B)^2 using the distributive property, we get:
C^2 = (A + B)(A + B) = A(A + B) + B(A + B).
- Using matrix multiplication rules, we can further simplify this expression:
C^2 = A^2 + AB + BA + B^2.
AB = BA
- Given that AB = BA, we can substitute this into the expression for C^2:
C^2 = A^2 + AB + BA + B^2 = A + AB + BA + B = C + C.
Proving C is Idempotent
- From the equation C^2 = C + C, we can rearrange to obtain:
C^2 - C - C = 0.
- Factoring out C, we have:
C(C - I) - C = 0,
C(C - I - 1) = 0,
C(C - 2I) = 0.
Determinant of C
- The determinant of a matrix represents the scaling factor of the matrix.
- If the determinant is zero, it means the matrix is singular and does not have an inverse.
Conclusion: Determinant of C
- From the equation C(C - 2I) = 0, we can conclude that either C or (C - 2I) must have a determinant of zero.
- Therefore, the determinant of C is zero, as one of the factors in the product is zero.
- An idempotent matrix is a square matrix that, when multiplied by itself, results in the same matrix.
- Mathematically, for a matrix A to be idempotent, it must satisfy the condition A^2 = A.
Sum of Idempotent Matrices
- Let A and B be two idempotent matrices.
- We need to show that the sum of A and B, denoted as C = A + B, is also idempotent.
- To prove this, we can calculate the square of C: C^2 = (A + B)^2.
Calculating (A + B)^2
- Expanding (A + B)^2 using the distributive property, we get:
C^2 = (A + B)(A + B) = A(A + B) + B(A + B).
- Using matrix multiplication rules, we can further simplify this expression:
C^2 = A^2 + AB + BA + B^2.
AB = BA
- Given that AB = BA, we can substitute this into the expression for C^2:
C^2 = A^2 + AB + BA + B^2 = A + AB + BA + B = C + C.
Proving C is Idempotent
- From the equation C^2 = C + C, we can rearrange to obtain:
C^2 - C - C = 0.
- Factoring out C, we have:
C(C - I) - C = 0,
C(C - I - 1) = 0,
C(C - 2I) = 0.
Determinant of C
- The determinant of a matrix represents the scaling factor of the matrix.
- If the determinant is zero, it means the matrix is singular and does not have an inverse.
Conclusion: Determinant of C
- From the equation C(C - 2I) = 0, we can conclude that either C or (C - 2I) must have a determinant of zero.
- Therefore, the determinant of C is zero, as one of the factors in the product is zero.
|
Explore Courses for Mathematics exam
|
|
Similar Mathematics Doubts
The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. Can you explain this answer?
Question Description
The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. 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 The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. Can you explain this answer?.
The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. 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 The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. Can you explain this answer?.
Solutions for The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.
Here you can find the meaning of The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. Can you explain this answer?, a detailed solution for The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. Can you explain this answer? has been provided alongside types of The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The sum of two idempotent matrices A and B is idempotent and if AB + BA = C, then what will be the determinant of C ?Correct answer is '0'. Can you explain this answer? tests, examples and also practice Mathematics tests.
|
|
Explore Courses for Mathematics exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup