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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.
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Totally based on properties of trace and eigen values

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).
We can take simple example with upper or lower triangular matrices. For options $(b) , (c)$ and $(d)$ reverse is always true.
by Active (3.8k points)
edited
–3
C can  also be true
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nope... not always(if two eigen values are negative than also determinant would be possive which contradict statement c)
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C may not be true bcz if u take -1,-2 ,3 eigen values so determinant = 6 positive but 2 eigen values are negative
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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.
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Good counter example

Taking a simple example we can get a clear idea.
Let matrix A be

 10 0 0 0 -2 0 0 0 -4

Here trace(A)=4 and determinant(A)=80.

As matrix is diagonal matrix the principal diagonal elements are the eigenvalues for matrix A.

Even though a trace of a matrix is positive we have two negative eigenvalues(i.e. -2,-4). So option B is false.

Even though determinant of a matrix is positive we have two negative eigenvalues(i.e. -2,-4). So option C is false.

the product of trace and determinant is 320 which is positive still we have two negative eigenvalues(i.e. -2,-4). So option D is false.

Now if the determinant is negative then at least one eigenvalue of the matrix is negative because the product of eigenvalues of a given matrix is equal to the determinant of a given matrix.
So OPTION A is True.

by Junior (717 points)
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Thank you bro😊
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Thank you
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I think knowing properties is better than solving by taking example.  :)