Let’s consider a 2x2 matrix P consisting of elements
a 0
0 b
We know that trace of a matrix = sum of its eigen values
and determinant of a matrix = product of its eigen values
hence, tr(P) = a+b and det(P) = ab holds true.
Now, option A says that
if the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative,
which means if trace is positive there are two ways :
1. either both a and b are positive or
2 one is positive and other is negative.
we are given that determinant is negative which means the product ab is negative. This happens only when one of either a and b is negative.
Hence option A is correct.
option B says that
If the trace of the matrix is positive, all its eigenvalues are positive.
As discussed above that it is not necessary that a and b are always positive. It’s because any one of a or b (which is greater) can be positive and other can be negative (the smaller value) and still the sum would result to positive. (Just like 9+(-2)). Also a good mention of real eigenvalues in the question supports this very fact.
Hence option B is not correct.
option C says that
If the determinant of the matrix is positive, all its eigenvalues are positive.
This is simple. if det(P) = ab is positive there can be two cases:
- both a and b are positive
- both a and b are negative.
Hence option C is not correct.
option D says that
If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.
If trace (P) * det (P) = ab * (a+b) is positive it means
- both trace and determinant are positive.
- both trace and determinant are negative.
First case is trivial one, let’s look onto second case
if trace (P) = a+b is negative then one of them is sure to be negative . Also if det (P) = ab is negative then one of a or b is negative which contradicts the above statement that all eigenvalues are positive.
Hence option D is not correct.
The crux is in the fact that the sum of the eigenvalues is the trace of the matrix and product of the eigenvalues is determinant of the matrix. Rest is simple thinking.
Hope it helps.