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If A is non — scalar, non — identity ivolutory matrix, then minimal polynomial mA(x) is
- a)x(x —1)
- b)(x—1 ) ( x + 1)
- c)x(1 —x)
- d)x + 1
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If A is non — scalar, non — identity ivolutory matrix, the...
Since A is idempotent A2 = A
A2 - A = 0
mA(x) = x2 - x
= x(x-1)
Most Upvoted Answer
If A is non — scalar, non — identity ivolutory matrix, the...
Since A is idempotent A2 = A
A2 - A = 0
mA(x) = x2 - x
= x(x-1)
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If A is non — scalar, non — identity ivolutory matrix, the...
Explanation:
Non-scalar, Non-identity Involuntary Matrix:
- A non-scalar matrix is a matrix that is not a multiple of the identity matrix.
- An involutory matrix is a square matrix that is its own inverse, i.e., A^2 = I.
- In this case, A is both non-scalar and involutory.
Minimal Polynomial:
- The minimal polynomial of a matrix A is the monic polynomial of least degree that has the matrix as a root.
- Since A is involutory, the minimal polynomial must divide x^2 - 1, as A^2 - I = 0.
Factors of x^2 - 1:
- The factors of x^2 - 1 are (x - 1) and (x + 1) by the difference of squares formula.
- The minimal polynomial of A must divide x^2 - 1, so it must be either (x - 1), (x + 1), or their product.
Correct Answer:
- Option B, (x - 1)(x + 1), is the correct answer because it includes both factors of x^2 - 1, making it a possible minimal polynomial for the given non-scalar, non-identity involutory matrix A.
Non-scalar, Non-identity Involuntary Matrix:
- A non-scalar matrix is a matrix that is not a multiple of the identity matrix.
- An involutory matrix is a square matrix that is its own inverse, i.e., A^2 = I.
- In this case, A is both non-scalar and involutory.
Minimal Polynomial:
- The minimal polynomial of a matrix A is the monic polynomial of least degree that has the matrix as a root.
- Since A is involutory, the minimal polynomial must divide x^2 - 1, as A^2 - I = 0.
Factors of x^2 - 1:
- The factors of x^2 - 1 are (x - 1) and (x + 1) by the difference of squares formula.
- The minimal polynomial of A must divide x^2 - 1, so it must be either (x - 1), (x + 1), or their product.
Correct Answer:
- Option B, (x - 1)(x + 1), is the correct answer because it includes both factors of x^2 - 1, making it a possible minimal polynomial for the given non-scalar, non-identity involutory matrix A.
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If A is non — scalar, non — identity ivolutory matrix, then minimal polynomial mA(x) isa)x(x —1)b)(x—1 ) ( x + 1)c)x(1 —x)d)x + 1Correct answer is option 'B'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about If A is non — scalar, non — identity ivolutory matrix, then minimal polynomial mA(x) isa)x(x —1)b)(x—1 ) ( x + 1)c)x(1 —x)d)x + 1Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If A is non — scalar, non — identity ivolutory matrix, then minimal polynomial mA(x) isa)x(x —1)b)(x—1 ) ( x + 1)c)x(1 —x)d)x + 1Correct answer is option 'B'. Can you explain this answer?.
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