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If A and B are square matrices of different order Cx(A) and cx(B) are characteristic polynomials of A and B respectively and it is given that Cx(B) is minimal polynomial of A as well then
- a)A is invertible implies B is invertible but B may be invertible even, if A is not so
- b)B is invertible implies A is invertible but A may be invertible even, if B is not so
- c)A and B are invertible together or fail together to be so
- d)No such relation is necessary
Correct answer is option 'C'. Can you explain this answer?
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If A and B are square matrices of different order Cx(A) and cx(B) are ...
Explanation:
To understand the given statement, let's break it down and analyze each part.
1. A and B are square matrices of different order:
This means that matrices A and B have different dimensions. Let's assume that matrix A is of order n x n and matrix B is of order m x m, where n ≠ m.
2. Cx(A) and Cx(B) are characteristic polynomials of A and B:
The characteristic polynomial of a square matrix is obtained by finding the determinant of the matrix minus a scalar multiple of the identity matrix. Let's assume that Cx(A) and Cx(B) are the characteristic polynomials of matrices A and B, respectively.
3. Cx(B) is the minimal polynomial of A:
The minimal polynomial of a matrix is the monic polynomial of least degree that annihilates the matrix. In other words, it is the smallest polynomial that when applied to the matrix gives the zero matrix. In this case, it is given that Cx(B) is the minimal polynomial of matrix A.
Now, let's analyze the given options:
a) A is invertible implies B is invertible but B may be invertible even if A is not so:
This option suggests that if matrix A is invertible, then matrix B must also be invertible. However, it also states that matrix B may be invertible even if matrix A is not. This statement is not necessarily true because the invertibility of a matrix is not solely dependent on the invertibility of another matrix.
b) B is invertible implies A is invertible but A may be invertible even if B is not so:
This option suggests that if matrix B is invertible, then matrix A must also be invertible. However, it also states that matrix A may be invertible even if matrix B is not. Similar to option (a), this statement is not necessarily true.
c) A and B are invertible together or fail together to be so:
This option suggests that matrices A and B are either both invertible or both not invertible. This statement is true because if matrix A is invertible, then its characteristic polynomial Cx(A) will not have any zero eigenvalues. Since Cx(B) is the minimal polynomial of A, it will also not have any zero eigenvalues. This implies that matrix B is also invertible. On the other hand, if matrix A is not invertible, then its characteristic polynomial Cx(A) will have at least one zero eigenvalue. Since Cx(B) is the minimal polynomial of A, it will also have at least one zero eigenvalue. This implies that matrix B is also not invertible.
d) No such relation is necessary:
This option suggests that there is no necessary relation between the invertibility of matrices A and B. However, as explained in option (c), there is indeed a relation between the invertibility of matrices A and B.
In conclusion:
The correct answer is option (c) - A and B are invertible together or fail together to be so. The invertibility of matrices A and B is related because their characteristic polynomials and the minimal polynomial of A are connected.
To understand the given statement, let's break it down and analyze each part.
1. A and B are square matrices of different order:
This means that matrices A and B have different dimensions. Let's assume that matrix A is of order n x n and matrix B is of order m x m, where n ≠ m.
2. Cx(A) and Cx(B) are characteristic polynomials of A and B:
The characteristic polynomial of a square matrix is obtained by finding the determinant of the matrix minus a scalar multiple of the identity matrix. Let's assume that Cx(A) and Cx(B) are the characteristic polynomials of matrices A and B, respectively.
3. Cx(B) is the minimal polynomial of A:
The minimal polynomial of a matrix is the monic polynomial of least degree that annihilates the matrix. In other words, it is the smallest polynomial that when applied to the matrix gives the zero matrix. In this case, it is given that Cx(B) is the minimal polynomial of matrix A.
Now, let's analyze the given options:
a) A is invertible implies B is invertible but B may be invertible even if A is not so:
This option suggests that if matrix A is invertible, then matrix B must also be invertible. However, it also states that matrix B may be invertible even if matrix A is not. This statement is not necessarily true because the invertibility of a matrix is not solely dependent on the invertibility of another matrix.
b) B is invertible implies A is invertible but A may be invertible even if B is not so:
This option suggests that if matrix B is invertible, then matrix A must also be invertible. However, it also states that matrix A may be invertible even if matrix B is not. Similar to option (a), this statement is not necessarily true.
c) A and B are invertible together or fail together to be so:
This option suggests that matrices A and B are either both invertible or both not invertible. This statement is true because if matrix A is invertible, then its characteristic polynomial Cx(A) will not have any zero eigenvalues. Since Cx(B) is the minimal polynomial of A, it will also not have any zero eigenvalues. This implies that matrix B is also invertible. On the other hand, if matrix A is not invertible, then its characteristic polynomial Cx(A) will have at least one zero eigenvalue. Since Cx(B) is the minimal polynomial of A, it will also have at least one zero eigenvalue. This implies that matrix B is also not invertible.
d) No such relation is necessary:
This option suggests that there is no necessary relation between the invertibility of matrices A and B. However, as explained in option (c), there is indeed a relation between the invertibility of matrices A and B.
In conclusion:
The correct answer is option (c) - A and B are invertible together or fail together to be so. The invertibility of matrices A and B is related because their characteristic polynomials and the minimal polynomial of A are connected.
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If A and B are square matrices of different order Cx(A) and cx(B) are characteristicpolynomials of A and B respectively and it is given that Cx(B) is minimal polynomial of A as well thena)A is invertible implies B is invertible but B may be invertible even, if A is not sob)B is invertible implies A is invertible but A may be invertible even, if B is not soc)A and B are invertible together or fail together to be sod)No such relation is necessaryCorrect answer is option 'C'. Can you explain this answer?
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If A and B are square matrices of different order Cx(A) and cx(B) are characteristicpolynomials of A and B respectively and it is given that Cx(B) is minimal polynomial of A as well thena)A is invertible implies B is invertible but B may be invertible even, if A is not sob)B is invertible implies A is invertible but A may be invertible even, if B is not soc)A and B are invertible together or fail together to be sod)No such relation is necessaryCorrect answer is option 'C'. 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 If A and B are square matrices of different order Cx(A) and cx(B) are characteristicpolynomials of A and B respectively and it is given that Cx(B) is minimal polynomial of A as well thena)A is invertible implies B is invertible but B may be invertible even, if A is not sob)B is invertible implies A is invertible but A may be invertible even, if B is not soc)A and B are invertible together or fail together to be sod)No such relation is necessaryCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If A and B are square matrices of different order Cx(A) and cx(B) are characteristicpolynomials of A and B respectively and it is given that Cx(B) is minimal polynomial of A as well thena)A is invertible implies B is invertible but B may be invertible even, if A is not sob)B is invertible implies A is invertible but A may be invertible even, if B is not soc)A and B are invertible together or fail together to be sod)No such relation is necessaryCorrect answer is option 'C'. Can you explain this answer?.
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