\(\displaystyle \begin{align*}&\text{A symmetric matrix is one that is equal to its transpose, i.e:}\\&A=A^{T}\\&\text{To review, a transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, etc.}\\&\text{Therefore, the symmetric matrix is:}\\&\begin{bmatrix}-7&16&-4&18&4&1\\16&-6&3&9&11&18\\-4&3&-17&3&-18&-3\\18&9&3&0&4&1\\4&11&-18&4&-4&-6\\1&18&-3&1&-6&16\end{bmatrix}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A symmetric matrix is one that is equal to its transpose, i.e:}\\&A=A^{T}\\&\text{To review, a transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, etc.}\\&\text{Therefore, the symmetric matrix is:}\\&\begin{bmatrix}-3&1&-4&19&8&-5&20\\1&-4&-5&5&-11&0&19\\-4&-5&15&-11&20&-18&14\\19&5&-11&6&3&-9&-4\\8&-11&20&3&1&6&-4\\-5&0&-18&-9&6&2&20\\20&19&14&-4&-4&20&-15\end{bmatrix}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A symmetric matrix is one that is equal to its transpose, i.e:}\\&A=A^{T}\\&\text{To review, a transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, etc.}\\&\text{Therefore, the symmetric matrix is:}\\&\begin{bmatrix}17&10&13&-18&10&5&3\\10&-14&-7&5&-7&8&-8\\13&-7&20&1&15&-20&-7\\-18&5&1&-15&9&16&-6\\10&-7&15&9&18&-11&3\\5&8&-20&16&-11&-3&11\\3&-8&-7&-6&3&11&20\end{bmatrix}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A symmetric matrix is one that is equal to its transpose, i.e:}\\&A=A^{T}\\&\text{To review, a transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, etc.}\\&\text{Therefore, the symmetric matrix is:}\\&\begin{bmatrix}-13&6&13&2&-17\\6&19&17&14&-7\\13&17&7&-10&-13\\2&14&-10&-6&-15\\-17&-7&-13&-15&4\end{bmatrix}\end{align*}\)

\(\displaystyle \begin{align*}&\text{A symmetric matrix is equal to its transpose, that is to say:}\\&A=A^{T}\\&\text{A transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, and so on.}\\&\text{The symmetric matrix is:}\\&\begin{bmatrix}14&6&-12&-3&-4&4\\6&17&17&-13&1&-4\\-12&17&-13&-20&4&-17\\-3&-13&-20&-2&16&9\\-4&1&4&16&-10&-13\\4&-4&-17&9&-13&-13\end{bmatrix}\end{align*}\)

\(\displaystyle \begin{align*}&\text{Rather than actually calculating eigenvalues, it is worth noting that symmetric matrices}\\&\text{always have real eigenvalues. A symmetric matrix is one that is equal to its transpose, i.e:}\\&A=A^{T}\\&\text{Therefore, the matrix which must have real eigenvalues is:}\\&\begin{bmatrix}-8&-17&3&15&-12&4\\-17&-9&6&-4&5&2\\3&6&13&6&-19&0\\15&-4&6&14&-8&-17\\-12&5&-19&-8&-6&-6\\4&2&0&-17&-6&9\end{bmatrix}\end{align*}\)

\(\displaystyle \begin{align*}&\text{Rather than actually calculating eigenvalues, it is worth noting that symmetric matrices}\\&\text{always have real eigenvalues. A symmetric matrix is one that is equal to its transpose, i.e:}\\&A=A^{T}\\&\text{Therefore, the matrix which must have real eigenvalues is:}\\&\begin{bmatrix}-2&-6&-14&8&-7&-5\\-6&20&-14&9&0&13\\-14&-14&-15&-14&7&-1\\8&9&-14&13&3&-11\\-7&0&7&3&-18&5\\-5&13&-1&-11&5&-6\end{bmatrix}\end{align*}\)

A square matrix \(\displaystyle A\) is defined to be skew-symmetric if its transpose \(\displaystyle A^{T}\) - the matrix resulting from interchanging its rows and its columns - is equal to its additive inverse; that is, if

\(\displaystyle A^{T} = -A\).

Interchanging rows and columns, we see that if 

\(\displaystyle A = \begin{bmatrix} 1& -3 &4 \\ 3& 1& 5\\ -4&-5 & 1\end{bmatrix}\),

then 

\(\displaystyle A^{T} = \begin{bmatrix} 1& 3 &-4 \\ -3& 1&-5\\ 4&5 & 1 \end{bmatrix}\).

\(\displaystyle -A\) can be determined by changing each element in \(\displaystyle A\) to its additive inverse:

\(\displaystyle A = \begin{bmatrix} -1& 3 &-4 \\- 3&- 1& -5\\ 4& 5 &- 1\end{bmatrix}\)

\(\displaystyle A^{T} \ne -A\), since not every element in corresponding positions is equal; in particular, the three elements in the main diagonal differ. \(\displaystyle A\) is not a skew-symmetric matrix.

A square matrix \(\displaystyle A\) is defined to be skew-symmetric if its transpose \(\displaystyle A^{T}\) - the matrix resulting from interchanging its rows and its columns - is equal to its additive inverse; that is, if

\(\displaystyle A^{T} = -A\).

Interchanging rows and columns, we see that if 

\(\displaystyle A = \begin{bmatrix} 0& -3 &4 \\ 3& 0& 5\\ -4&-5 & 0 \end{bmatrix}\),

then 

\(\displaystyle A^{T} = \begin{bmatrix} 0& 3 &-4 \\ -3& 0&-5\\ 4&5 & 0 \end{bmatrix}\).

We see that each element of \(\displaystyle A^{T}\) is the additive inverse of the corresponding element in \(\displaystyle A\), so \(\displaystyle A^{T} = -A\), and \(\displaystyle A\) is skew-symmetric.

A square matrix \(\displaystyle A\) is defined to be skew-symmetric if its transpose \(\displaystyle A^{T}\) - the matrix resulting from interchanging its rows and its columns - is equal to its additive inverse; that is, if

\(\displaystyle A^{T} = -A\).

Therefore, by substitution, 

\(\displaystyle A^{T}A^{-1} = (-A)A^{-1}\)

\(\displaystyle A^{T}A^{-1} = (-1 A)A^{-1}\)

\(\displaystyle A^{T}A^{-1} = -1(AA^{-1})\)

\(\displaystyle A^{T}A^{-1} = -1I\)

\(\displaystyle A^{T}A^{-1}\) must be equal to the opposite of the three-by-three identity matrix, which is \(\displaystyle \begin{bmatrix} -1& 0& 0\\ 0&-1 &0 \\ 0&0 &-1 \end{bmatrix}\).