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GATE 2014 [Set-3]
April 4, 2025GATE 2014 [Set-3]
April 4, 2025GATE 2014 [Set-3]
| Question 14 |
Which one of the following statements is TRUE about every n × n matrix with only real eigenvalues?
| If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative. | |
| If the trace of the matrix is positive, all its eigenvalues are positive. | |
| If the determinant of the matrix is positive, all its eigenvalues are positive. | |
| If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive. |
Question 14 Explanation:
The sum of the n eigenvalues of A is the same as the trace of A (that is, the sum of the diagonal
elements of A).
• The product of the n eigenvalues of A is the same as the determinant of A. •
A: Yes, for sum to be negative there should be atleast one negative number.
B: There can be one small negative number and remaining positive, where sum is positive.
C: Product of two negative numbers is positive. So, there no need of all positive eigen values.
D: There is no need for all eigen values to be positive, as product of two negative numbers is
positive.
elements of A).
• The product of the n eigenvalues of A is the same as the determinant of A. •
A: Yes, for sum to be negative there should be atleast one negative number.
B: There can be one small negative number and remaining positive, where sum is positive.
C: Product of two negative numbers is positive. So, there no need of all positive eigen values.
D: There is no need for all eigen values to be positive, as product of two negative numbers is
positive.
Correct Answer: A
Question 14 Explanation:
The sum of the n eigenvalues of A is the same as the trace of A (that is, the sum of the diagonal
elements of A).
• The product of the n eigenvalues of A is the same as the determinant of A. •
A: Yes, for sum to be negative there should be atleast one negative number.
B: There can be one small negative number and remaining positive, where sum is positive.
C: Product of two negative numbers is positive. So, there no need of all positive eigen values.
D: There is no need for all eigen values to be positive, as product of two negative numbers is
positive.
elements of A).
• The product of the n eigenvalues of A is the same as the determinant of A. •
A: Yes, for sum to be negative there should be atleast one negative number.
B: There can be one small negative number and remaining positive, where sum is positive.
C: Product of two negative numbers is positive. So, there no need of all positive eigen values.
D: There is no need for all eigen values to be positive, as product of two negative numbers is
positive.