Symmetric Matrices

Practice Questions

Linear Algebra › Symmetric Matrices

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1

$\begin{align*}&\text{Decide which of the given matrices must have real eigenvalues, without calculating eigenvalues or vectors.}\end{align*}$

2

$\begin{align*}&\text{Select the symmetric matrix from the following choices:}\end{align*}$

3

$\begin{align*}&\text{Choose the symmetric matrix from the four choices given:}\end{align*}$

4

$A = \begin{bmatrix} 0 & 6i & 3i \ -6i & 0 & 2i \ -3i & -2i & 0 \end{bmatrix}$

Is is an example of a symmetric matrix, a skew-symmetric matrix, or a Hermitian matrix?

5

$A= \begin{bmatrix} 0 & \cos \frac{5\pi }{7} + i \sin \frac{5\pi }{7} \ \ \cos \frac{12\pi }{7} + i \sin \frac{12\pi }{7} & 0 \end{bmatrix}$

Is is an example of a symmetric matrix, a skew-symmetric matrix, or a Hermitian matrix?

6

$A^{T} = \begin{bmatrix} 2+ i & 5 \ 3 - 6i & 2i \ 1 + 7i & 0 \end{bmatrix}$

Which of the following is equal to $A^{*}$

7

True or False: All skew-symmetric matrices are also symmetric matrices.

8

$\begin{align*}&\text{Decide which of the given matrices must have real eigenvalues, without calculating eigenvalues or vectors.}\end{align*}$

9

$A = \begin{bmatrix} 0 & 4 - i\sqrt{2} \ x & 0\end{bmatrix}$

Evaluate so that is a skew-Hermitian matrix.

10

Which of the following dimensions cannot be that of a symmetric matrix?

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