Which of the following statement is correct?a)The matrix A and its tra...
Explanation:
Characteristic polynomial:
The characteristic polynomial of a square matrix A is defined as the polynomial det(A - λI) where det denotes the determinant, I is the identity matrix, and λ is a scalar variable.
Statement (a):
The matrix A and its transpose AT have the same characteristic polynomial.
This statement is true. The proof is given below:
- Let A be a square matrix of order n.
- Then, the characteristic polynomial of A is det(A - λI) where I is the n × n identity matrix.
- Now, consider the transpose of A, denoted by AT.
- The characteristic polynomial of AT is det(AT - λI).
- We know that (AT - λI) = (A - λI)T.
- Therefore, det(AT - λI) = det((A - λI)T) = det(A - λI).
- Hence, the matrix A and its transpose AT have the same characteristic polynomial.
Statement (b):
Similar matrices have not the same characteristic polynomial.
This statement is false. Similar matrices have the same characteristic polynomial.
- Two matrices A and B are said to be similar if there exists an invertible matrix P such that B = P-1AP.
- If A and B are similar matrices, then they have the same eigenvalues.
- The eigenvalues of a matrix are the roots of its characteristic polynomial.
- Hence, similar matrices have the same characteristic polynomial.
Statement (c):
The trace of two similar matrices are not same.
This statement is false. Similar matrices have the same trace.
- The trace of a matrix is the sum of its diagonal elements.
- If A and B are similar matrices, then they have the same characteristic polynomial and hence the same eigenvalues.
- The sum of the eigenvalues of a matrix is equal to its trace.
- Therefore, similar matrices have the same trace.
Statement (d):
The determinant of two similar matrices is not same.
This statement is false. Similar matrices have the same determinant.
- If A and B are similar matrices, then they have the same characteristic polynomial and hence the same eigenvalues.
- The determinant of a matrix is equal to the product of its eigenvalues.
- Therefore, similar matrices have the same determinant.