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If A is non - scalar, non - identity idempotent matrix of order n ≥ 2. Then, minimal polynomial mA(x) is
- a)x ( x -1)
- b)x (x + 1 )
- c)x ( 1 - x)
- d)x 2( 1+ x)
Correct answer is option 'A'. Can you explain this answer?
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If A is non - scalar, non - identity idempotent matrix of order n ≥...
If A is a non-scalar, non-identity idempotent matrix of order n, then it means that A is a square matrix of size n × n that is not a scalar multiple of the identity matrix (I) and satisfies the idempotent property.
The idempotent property means that when A is multiplied by itself, the result is still A. Mathematically, this can be represented as A^2 = A.
Since A is not the identity matrix, there exists at least one element in A that is not equal to 1 on the main diagonal. Let's say this element is a_ij.
Now, when we calculate A^2, each element of the resulting matrix is obtained by taking the dot product of the corresponding row of A with the corresponding column of A.
For the element in the i-th row and j-th column of A^2, denoted as (A^2)_ij, we have:
(A^2)_ij = a_i1 * a_1j + a_i2 * a_2j + ... + a_in * a_nj
Since A is idempotent, we know that (A^2)_ij = a_ij. Therefore, we have:
a_ij = a_i1 * a_1j + a_i2 * a_2j + ... + a_in * a_nj
From this equation, we can see that a_ij must be equal to the sum of the products of the elements in the i-th row of A multiplied by the corresponding elements in the j-th column of A.
Since A is not a scalar multiple of the identity matrix, there must exist at least one element a_ij that is not equal to 0. This means that at least one row of A must have more than one non-zero element.
Therefore, we can conclude that if A is a non-scalar, non-identity idempotent matrix of order n, then it must have at least one row with more than one non-zero element.
The idempotent property means that when A is multiplied by itself, the result is still A. Mathematically, this can be represented as A^2 = A.
Since A is not the identity matrix, there exists at least one element in A that is not equal to 1 on the main diagonal. Let's say this element is a_ij.
Now, when we calculate A^2, each element of the resulting matrix is obtained by taking the dot product of the corresponding row of A with the corresponding column of A.
For the element in the i-th row and j-th column of A^2, denoted as (A^2)_ij, we have:
(A^2)_ij = a_i1 * a_1j + a_i2 * a_2j + ... + a_in * a_nj
Since A is idempotent, we know that (A^2)_ij = a_ij. Therefore, we have:
a_ij = a_i1 * a_1j + a_i2 * a_2j + ... + a_in * a_nj
From this equation, we can see that a_ij must be equal to the sum of the products of the elements in the i-th row of A multiplied by the corresponding elements in the j-th column of A.
Since A is not a scalar multiple of the identity matrix, there must exist at least one element a_ij that is not equal to 0. This means that at least one row of A must have more than one non-zero element.
Therefore, we can conclude that if A is a non-scalar, non-identity idempotent matrix of order n, then it must have at least one row with more than one non-zero element.
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If A is non - scalar, non - identity idempotent matrix of order n ≥ 2. Then, minimal polynomial mA(x) isa)x ( x -1)b)x (x + 1 )c)x ( 1 - x)d)x 2( 1+ x)Correct answer is option 'A'. 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 idempotent matrix of order n ≥ 2. Then, minimal polynomial mA(x) isa)x ( x -1)b)x (x + 1 )c)x ( 1 - x)d)x 2( 1+ x)Correct answer is option 'A'. 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 idempotent matrix of order n ≥ 2. Then, minimal polynomial mA(x) isa)x ( x -1)b)x (x + 1 )c)x ( 1 - x)d)x 2( 1+ x)Correct answer is option 'A'. Can you explain this answer?.
If A is non - scalar, non - identity idempotent matrix of order n ≥ 2. Then, minimal polynomial mA(x) isa)x ( x -1)b)x (x + 1 )c)x ( 1 - x)d)x 2( 1+ x)Correct answer is option 'A'. 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 idempotent matrix of order n ≥ 2. Then, minimal polynomial mA(x) isa)x ( x -1)b)x (x + 1 )c)x ( 1 - x)d)x 2( 1+ x)Correct answer is option 'A'. 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 idempotent matrix of order n ≥ 2. Then, minimal polynomial mA(x) isa)x ( x -1)b)x (x + 1 )c)x ( 1 - x)d)x 2( 1+ x)Correct answer is option 'A'. Can you explain this answer?.
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