Question

Which one of the following statements is NOT true for a square matrix A?
1. If A is upper triangular, the eigenvalues of A are the diagonal elements of it
2. If A is real symmetric, the eigenvalues of A are always real and positive
3. If A is real, the eigenvalues of A and A^T are always the same
4. If all the principal minors of A are positive, all the eigenvalues of A are also positive

Answer 1

Correct Answer - Option 2 : If A is real symmetric, the eigenvalues of A are always real and positive

Analyzing option A:

Let an upper triangular Matrix be:

\(A = \left[ {\begin{array}{*{20}{c}} 1&3\\ 0&2 \end{array}} \right]\) 

\(\left[ {A - \lambda I} \right] = \left[ {\begin{array}{*{20}{c}} {1 - \lambda }&3\\ 0&{2 - \lambda } \end{array}} \right]\) 

Eigen values: |A - λI| = 0

(1 - λ) (2 - λ) = 0

λ = 1, 2

Clearly eigen values are same as diagonal elements of matrix A.

Hence, A is True.

Analyzing option B:

Let a real symmetric matrix be:

\(A = \left[ {\begin{array}{*{20}{c}} { - 1}&0\\ 0&{ - 1} \end{array}} \right]\) 

\(\left[ {A - \lambda I} \right] = \left[ {\begin{array}{*{20}{c}} { - 1 - \lambda }&0\\ 0&{ - 1 - \lambda } \end{array}} \right]\) 

|A - λI| = (-1 - λ)² = (1 + λ)²

Eigen values are |A – λI| = 0

(1 + λ)² = 0

λ = -1, -1

Clearly, the eigenvalues are real but not positive. So, for a real symmetric matrix, the eigenvalues are real but need not be positive.

Hence option B is not true.

Remember:

1) Eigenvalues of A & A^T are the same.

2) But eigenvectors of A & A^T are not the same.

3) If all the principal minors of A are positive, all the eigenvalues of A are also positive.