If A is upper triangular, the eigen values of A are the diagonal elements of it
If A is real symmetric, the eigen values of A are always real and positive
If A is real, the eigen values of A and AT are always the same
If all the principal minors of A are positive, all the eigen values of A are also positive
Answer is Right!
Answer is Wrong!
The correct answer is D. If all the principal minors of A are positive, all the eigen values of A are also positive.
A principal minor of a square matrix is the determinant of a submatrix of the matrix. The principal minors of a square matrix are always non-negative. If all the principal minors of a square matrix are positive, then the matrix is said to be positive definite. A positive definite matrix has all positive eigenvalues.
However, there are square matrices that have all positive principal minors but negative eigenvalues. For example, the matrix $$A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}$$ has all positive principal minors but eigenvalues $3$ and $-1$.
Therefore, statement D is not always true.