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Which one of the following statements is TRUE about every $n \times n$ matrix with only real eigenvalues?

  1. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.
  2. If the trace of the matrix is positive, all its eigenvalues are positive.
  3. If the determinant of the matrix is positive, all its eigenvalues are positive.
  4. If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.
asked in Linear Algebra by Veteran (106k points) | 1.6k views

1 Answer

+27 votes
Best answer
Trace is the sum of all diagonal elements of a square matrix.
Determinant of a matrix = Product of eigen values.

$A)$ Is the right answer. To have the determinant negative ,at least one eigen value has to be negative(but reverse may not be true).
{you can take simple example with upper or lower triangular matrices. In the case option $(b) , $(c)$ and $(d)$ reverse is always true .}
answered by Active (3.7k points)
edited by
–3
C can  also be true
0
nope... not always(if two eigen values are negative than also determinant would be possive which contradict statement c)
+1
C may not be true bcz if u take -1,-2 ,3 eigen values so determinant = 6 positive but 2 eigen values are negative
+1
Counter ex for (B) is, Trace = 1 but eigen values are 2,-1 .

Counter ex for (C) is, Det = 2 but eigen values are -2,-1 .

Counter ex for (D) is,Trace * Det = 2 but eigen values =1,-2.
Answer:

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