$\displaystyle A$ is a skew-Hermitian matrix if it is equal to the additive inverse of its conjugate transpose - that is, if

$\displaystyle A = -A^{*}$

To determine whether this is the case, first, find the transpose of $\displaystyle A$ by exchanging rows with columns in $\displaystyle A$:

$\displaystyle A= \begin{bmatrix} 0 & 8+7 i \\ -8+7i &0 \end{bmatrix}$

$\displaystyle A^{T}= \begin{bmatrix} 0 & -8+7 i \\ 8+7i &0 \end{bmatrix}$

Obtain the conjugate transpose by changing each element to its complex conjugate:

$\displaystyle A^{*}= \begin{bmatrix} 0 & -8-7 i \\ 8-7i &0 \end{bmatrix}$

Now find the additive inverse of this by changing each entry to its additive inverse:

$\displaystyle -A^{*}= \begin{bmatrix} 0 & 8+7 i \\- 8+7i &0 \end{bmatrix} = A$

$\displaystyle A = -A^{*}$, so $\displaystyle A$ is skew-Hermitian.

$\displaystyle A$ is a skew-Hermitian matrix if it is equal to the additive inverse of its conjugate transpose - that is, if

$\displaystyle A = -A^{*}$

To determine whether this is the case, first, find the transpose of $\displaystyle A$ by exchanging rows with columns in $\displaystyle A$:

$\displaystyle A= \begin{bmatrix} 1 & 8+7 i \\ -8+7i &1 \end{bmatrix}$

$\displaystyle A^{T}= \begin{bmatrix}1 & -8+7 i \\ 8+7i &1 \end{bmatrix}$

Obtain the conjugate transpose by changing each element to its complex conjugate:

$\displaystyle A^{*}= \begin{bmatrix} 1 & -8-7 i \\ 8-7i &1\end{bmatrix}$

Now find the additive inverse of this by changing each entry to its additive inverse:

$\displaystyle -A^{*}= \begin{bmatrix} -1 & 8+7 i \\- 8+7i &-1 \end{bmatrix} \ne A$

$\displaystyle A \ne -A^{*}$, so $\displaystyle A$ is not skew-Hermitian.

$\displaystyle A$ is a skew-Hermitian matrix if it is equal to the additive inverse of its conjugate transpose - that is, if

$\displaystyle A = -A^{*}$

To determine whether this is the case, first, find the transpose of $\displaystyle A$ by exchanging rows with columns in $\displaystyle A$:

$\displaystyle A= \begin{bmatrix} 1 & 3+ i & -2 - i \\ -3+i &1 & 5i \\ 2-i &5i & 1\end{bmatrix}$

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

Obtain the conjugate transpose by changing each element to its complex conjugate:

$\displaystyle A^{*}= \begin{bmatrix} 1 &- 3-i & 2+i \\ 3- i &1 & -5i \\ -2 + i &-5i & 1\end{bmatrix}$

Now find the additive inverse of this by changing each entry to its additive inverse:

$\displaystyle -A^{*} = \begin{bmatrix} -1 & 3+ i & -2 - i \\ -3+i &-1 & 5i \\ 2-i &5i & -1\end{bmatrix}$

$\displaystyle A \ne -A^{*}$, so $\displaystyle A$ is not skew-Hermitian.

$\displaystyle A$ is a skew-Hermitian matrix if it is equal to the additive inverse of its conjugate transpose - that is, if

$\displaystyle A = -A^{*}$

To determine whether this is the case, first, find the transpose of $\displaystyle A$ by exchanging rows with columns in $\displaystyle A$:

$\displaystyle A= \begin{bmatrix} 0& 3+ i & -2 - i \\ -3+i &0 & 5i \\ 2-i &5i &0\end{bmatrix}$

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

Obtain the conjugate transpose by changing each element to its complex conjugate:

$\displaystyle A^{*}= \begin{bmatrix} 0 &- 3-i & 2+i \\ 3- i &0 & -5i \\ -2 + i &-5i &0\end{bmatrix}$

Now find the additive inverse of this by changing each entry to its additive inverse:

$\displaystyle -A^{*} = \begin{bmatrix} 0& 3+ i & -2 - i \\ -3+i &0 & 5i \\ 2-i &5i & 0\end{bmatrix} = A$

$\displaystyle A = -A^{*}$, so $\displaystyle A$ is skew-Hermitian.

$\displaystyle A$ is a skew-Hermitian matrix if it is equal to the additive inverse of its conjugate transpose - that is, if

$\displaystyle A = -A^{*}$

Therefore, first, take the transpose of $\displaystyle A$:

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

Obtain the conjugate transpose by changing each element to its complex conjugate:

$\displaystyle A ^{*}= \begin{bmatrix} 0 &\overline{x} \\ 4+ i\sqrt{2} & 0\end{bmatrix}$

Now find the additive inverse of this by changing each entry to its additive inverse:

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

For $\displaystyle A = -A^{*}$, or,

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

It is necessary and sufficient that the two equations

$\displaystyle - \overline{x} = 4-i\sqrt{2}$

and

$\displaystyle x =-4- i\sqrt{2}$

These conditions are equivalent, so

$\displaystyle x =-4- i\sqrt{2}$

makes $\displaystyle A$ skew-Hermitian.

A skew-symmetric matrix $\displaystyle A$ is one that becomes negative once the transpose is taken, or $\displaystyle A^T=-A$.

We have

$\displaystyle \begin{bmatrix} 0&-5 \\ 5& 0 \end{bmatrix}^T =\begin{bmatrix} 0& 5\\ -5& 0 \end{bmatrix} = -\begin{bmatrix} 0&-5 \\ 5& 0 \end{bmatrix}$.

Hence $\displaystyle \begin{bmatrix} 0&-5 \\ 5& 0 \end{bmatrix}$ is skew-symmetric.