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Let A be area 4 x 4 matrix with characteristic polynomial C(x) = (x2 + 1)2 which of the following is true?
- a)A is diagonalzable over complex number but not over real numbers.
- b)A is nilpotent
- c)A is invertible
- d)there is no such matrix A
Correct answer is option 'C'. Can you explain this answer?
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Let A be area 4 x 4 matrix with characteristic polynomial C(x) = (x2 +...
Explanation:
To determine which of the given options is true, we need to analyze the information provided in the question.
Characteristic Polynomial:
The characteristic polynomial of a matrix is obtained by subtracting the variable x from the main diagonal of the matrix and taking the determinant. In this case, the characteristic polynomial is given as C(x) = (x^2 - 1)^2.
Diagonalizable:
A matrix is said to be diagonalizable if it can be written in the form PDP^-1, where P is an invertible matrix and D is a diagonal matrix. In other words, if a matrix has n linearly independent eigenvectors, it is diagonalizable.
Option a: A is diagonalizable over complex numbers but not over real numbers.
This option cannot be true because if A is diagonalizable over complex numbers, it must also be diagonalizable over real numbers.
Option b: A is nilpotent.
A matrix is nilpotent if there exists a positive integer k such that A^k = 0. Since the characteristic polynomial of A is not of the form x^n, where n is a positive integer, A cannot be nilpotent.
Option c: A is invertible.
To determine if A is invertible, we need to check if the characteristic polynomial has any roots. The characteristic polynomial (x^2 - 1)^2 = 0 has two distinct roots, x = 1 and x = -1. Therefore, A has distinct eigenvalues, which implies that A is invertible.
Option d: There is no such matrix A.
This option cannot be true because the characteristic polynomial (x^2 - 1)^2 can correspond to a 4x4 matrix.
Therefore, the correct answer is option c) A is invertible.
To determine which of the given options is true, we need to analyze the information provided in the question.
Characteristic Polynomial:
The characteristic polynomial of a matrix is obtained by subtracting the variable x from the main diagonal of the matrix and taking the determinant. In this case, the characteristic polynomial is given as C(x) = (x^2 - 1)^2.
Diagonalizable:
A matrix is said to be diagonalizable if it can be written in the form PDP^-1, where P is an invertible matrix and D is a diagonal matrix. In other words, if a matrix has n linearly independent eigenvectors, it is diagonalizable.
Option a: A is diagonalizable over complex numbers but not over real numbers.
This option cannot be true because if A is diagonalizable over complex numbers, it must also be diagonalizable over real numbers.
Option b: A is nilpotent.
A matrix is nilpotent if there exists a positive integer k such that A^k = 0. Since the characteristic polynomial of A is not of the form x^n, where n is a positive integer, A cannot be nilpotent.
Option c: A is invertible.
To determine if A is invertible, we need to check if the characteristic polynomial has any roots. The characteristic polynomial (x^2 - 1)^2 = 0 has two distinct roots, x = 1 and x = -1. Therefore, A has distinct eigenvalues, which implies that A is invertible.
Option d: There is no such matrix A.
This option cannot be true because the characteristic polynomial (x^2 - 1)^2 can correspond to a 4x4 matrix.
Therefore, the correct answer is option c) A is invertible.
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