Question

Which one of the following statements TRUE about evevy n x n matrix with only real eigen values?

• A. If the trace of the matrix is positive and the determinant is negative, at least one of its eigen values in negative
• B. If the trace of the matrix is positive, all eigen values are positive.
• C. If the determinant of the matrix is positive, all its eigen values are positive.
• D. If the product ot the trace and determinant of the matrix is positive, all its eigen values are positive.

Answer 1

The correct option is A If the trace of the matrix is positive and the determinant is negative, at least one of its eigen values in negative

$\therefore$ Determinant of a matrix = product of its eigen values.
So, determinant is negative $\Rightarrow$ there exists at least one eigen value, which is negative.