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Linear Algebra : Symmetric Matrices

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Example Questions

Example Question #1 : Symmetric Matrices

Which matrix is symmetric?

Possible Answers:

$\begin{bmatrix} 1 &2 &1 \\2 &1 &5 \\1 &-5 &1 \end{bmatrix}$ $\displaystyle \begin{bmatrix} 1 &2 &1 \\2 &1 &5 \\1 &-5 &1 \end{bmatrix}$

$\begin{bmatrix} 1 &3 &-2 \\ 3& -2& 1\end{bmatrix}$ $\displaystyle \begin{bmatrix} 1 &3 &-2 \\ 3& -2& 1\end{bmatrix}$

$\begin{bmatrix} 0 &1 \\0 &1 \end{bmatrix}$ $\displaystyle \begin{bmatrix} 0 &1 \\0 &1 \end{bmatrix}$

$\begin{bmatrix} 1 &3 &-2 \\ 3& 6& 5\\-2 &5 & 0\end{bmatrix}$ $\displaystyle \begin{bmatrix} 1 &3 &-2 \\ 3& 6& 5\\-2 &5 & 0\end{bmatrix}$

Correct answer:

$\begin{bmatrix} 1 &3 &-2 \\ 3& 6& 5\\-2 &5 & 0\end{bmatrix}$ $\displaystyle \begin{bmatrix} 1 &3 &-2 \\ 3& 6& 5\\-2 &5 & 0\end{bmatrix}$

Explanation:

A symmetric matrix is symmetrical across the main diagonal. The numbers in the main diagonal can be anything, but the numbers in corresponding places on either side must be the same. In the correct answer, the matching numbers are the 3's, the -2's, and the 5's.

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Example Question #2 : Symmetric Matrices

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

Possible Answers:

$\begin{bmatrix}-3&13&23\\18&11&3\\19&9&-3\end{bmatrix}$ $\displaystyle \begin{bmatrix}-3&13&23\\18&11&3\\19&9&-3\end{bmatrix}$

$\begin{bmatrix}11&15&4\\23&12&-24\\12&-17&-12\end{bmatrix}$ $\displaystyle \begin{bmatrix}11&15&4\\23&12&-24\\12&-17&-12\end{bmatrix}$

$\begin{bmatrix}13&7&-18\\5&2&8\\-19&9&15\end{bmatrix}$ $\displaystyle \begin{bmatrix}13&7&-18\\5&2&8\\-19&9&15\end{bmatrix}$

$\begin{bmatrix}-18&8&20\\8&-11&14\\20&14&3\end{bmatrix}$ $\displaystyle \begin{bmatrix}-18&8&20\\8&-11&14\\20&14&3\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-18&8&20\\8&-11&14\\20&14&3\end{bmatrix}$ $\displaystyle \begin{bmatrix}-18&8&20\\8&-11&14\\20&14&3\end{bmatrix}$

Explanation:

$\begin{align*}&\text{A symmetric matrix is equal to its transpose, that is to say:}\\&A=A^{T}\\&\text{A transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, and so on.}\\&\text{The symmetric matrix is:}\\&\begin{bmatrix}-18&8&20\\8&-11&14\\20&14&3\end{bmatrix}\end{align*}$ $\displaystyle \begin{align*}&\text{A symmetric matrix is equal to its transpose, that is to say:}\\&A=A^{T}\\&\text{A transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, and so on.}\\&\text{The symmetric matrix is:}\\&\begin{bmatrix}-18&8&20\\8&-11&14\\20&14&3\end{bmatrix}\end{align*}$

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Example Question #1 : Symmetric Matrices

$\begin{align*}&\text{Which of the following matrices is symmetric?}\end{align*}$ $\displaystyle \begin{align*}&\text{Which of the following matrices is symmetric?}\end{align*}$

Possible Answers:

$\begin{bmatrix}15&6&15\\20&-16&20\\-13&-6&-13\end{bmatrix}$ $\displaystyle \begin{bmatrix}15&6&15\\20&-16&20\\-13&-6&-13\end{bmatrix}$

$\begin{bmatrix}-17&-36&-20\\-27&13&5\\-13&-2&-5\end{bmatrix}$ $\displaystyle \begin{bmatrix}-17&-36&-20\\-27&13&5\\-13&-2&-5\end{bmatrix}$

$\begin{bmatrix}-1&-13&-7\\-18&16&8\\-1&-13&-7\end{bmatrix}$ $\displaystyle \begin{bmatrix}-1&-13&-7\\-18&16&8\\-1&-13&-7\end{bmatrix}$

$\begin{bmatrix}-19&-7&14\\-7&-7&2\\14&2&1\end{bmatrix}$ $\displaystyle \begin{bmatrix}-19&-7&14\\-7&-7&2\\14&2&1\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-19&-7&14\\-7&-7&2\\14&2&1\end{bmatrix}$ $\displaystyle \begin{bmatrix}-19&-7&14\\-7&-7&2\\14&2&1\end{bmatrix}$

Explanation:

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Example Question #2 : Symmetric Matrices

$\begin{align*}&\text{Which of the following matrices is symmetric?}\end{align*}$ $\displaystyle \begin{align*}&\text{Which of the following matrices is symmetric?}\end{align*}$

Possible Answers:

$\begin{bmatrix}-12&-16&-18&11&11&13&-20&1\\-16&-18&10&-1&-18&18&-16&6\\-18&10&4&-17&2&-14&-11&-16\\11&-1&-17&-1&-19&5&8&-20\\11&-18&2&-19&-13&8&5&-4\\13&18&-14&5&8&10&-19&19\\-20&-16&-11&8&5&-19&1&-3\\1&6&-16&-20&-4&19&-3&15\end{bmatrix}$ $\displaystyle \begin{bmatrix}-12&-16&-18&11&11&13&-20&1\\-16&-18&10&-1&-18&18&-16&6\\-18&10&4&-17&2&-14&-11&-16\\11&-1&-17&-1&-19&5&8&-20\\11&-18&2&-19&-13&8&5&-4\\13&18&-14&5&8&10&-19&19\\-20&-16&-11&8&5&-19&1&-3\\1&6&-16&-20&-4&19&-3&15\end{bmatrix}$

$\begin{bmatrix}6&12&10&7&-4&12&18&5\\11&7&13&16&19&13&-7&-13\\-4&-14&8&4&-3&-3&-9&18\\-16&-16&-7&16&-12&2&20&-17\\-16&-16&-7&16&-12&2&20&-17\\-4&-14&8&4&-3&-3&-9&18\\11&7&13&16&19&13&-7&-13\\6&12&10&7&-4&12&18&5\end{bmatrix}$ $\displaystyle \begin{bmatrix}6&12&10&7&-4&12&18&5\\11&7&13&16&19&13&-7&-13\\-4&-14&8&4&-3&-3&-9&18\\-16&-16&-7&16&-12&2&20&-17\\-16&-16&-7&16&-12&2&20&-17\\-4&-14&8&4&-3&-3&-9&18\\11&7&13&16&19&13&-7&-13\\6&12&10&7&-4&12&18&5\end{bmatrix}$

$\begin{bmatrix}-9&9&16&-16&1&-7&3&28\\0&12&2&-4&-5&-7&-5&-8\\7&10&-4&-7&-17&-6&-5&-9\\-9&3&2&-18&15&27&-1&8\\-7&4&-8&6&25&-12&-10&-4\\1&1&1&18&-21&-6&23&5\\12&-14&4&-10&-2&14&-11&-35\\21&1&-18&1&-12&-4&-28&29\end{bmatrix}$ $\displaystyle \begin{bmatrix}-9&9&16&-16&1&-7&3&28\\0&12&2&-4&-5&-7&-5&-8\\7&10&-4&-7&-17&-6&-5&-9\\-9&3&2&-18&15&27&-1&8\\-7&4&-8&6&25&-12&-10&-4\\1&1&1&18&-21&-6&23&5\\12&-14&4&-10&-2&14&-11&-35\\21&1&-18&1&-12&-4&-28&29\end{bmatrix}$

$\begin{bmatrix}8&7&-8&1&1&-8&7&8\\3&6&-15&-17&-17&-15&6&3\\12&-17&-16&6&6&-16&-17&12\\6&-15&-8&-3&-3&-8&-15&6\\8&-14&-8&-17&-17&-8&-14&8\\-2&-7&-3&-6&-6&-3&-7&-2\\-10&2&10&2&2&10&2&-10\\-6&-2&14&13&13&14&-2&-6\end{bmatrix}$ $\displaystyle \begin{bmatrix}8&7&-8&1&1&-8&7&8\\3&6&-15&-17&-17&-15&6&3\\12&-17&-16&6&6&-16&-17&12\\6&-15&-8&-3&-3&-8&-15&6\\8&-14&-8&-17&-17&-8&-14&8\\-2&-7&-3&-6&-6&-3&-7&-2\\-10&2&10&2&2&10&2&-10\\-6&-2&14&13&13&14&-2&-6\end{bmatrix}$

Correct answer:

Explanation:

$\begin{align*}&\text{A symmetric matrix is one that is equal to its transpose, i.e:}\\&A=A^{T}\\&\text{To review, a transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, etc.}\\&\text{Therefore, the symmetric matrix is:}\\&\begin{bmatrix}-12&-16&-18&11&11&13&-20&1\\-16&-18&10&-1&-18&18&-16&6\\-18&10&4&-17&2&-14&-11&-16\\11&-1&-17&-1&-19&5&8&-20\\11&-18&2&-19&-13&8&5&-4\\13&18&-14&5&8&10&-19&19\\-20&-16&-11&8&5&-19&1&-3\\1&6&-16&-20&-4&19&-3&15\end{bmatrix}\end{align*}$ $\displaystyle \begin{align*}&\text{A symmetric matrix is one that is equal to its transpose, i.e:}\\&A=A^{T}\\&\text{To review, a transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, etc.}\\&\text{Therefore, the symmetric matrix is:}\\&\begin{bmatrix}-12&-16&-18&11&11&13&-20&1\\-16&-18&10&-1&-18&18&-16&6\\-18&10&4&-17&2&-14&-11&-16\\11&-1&-17&-1&-19&5&8&-20\\11&-18&2&-19&-13&8&5&-4\\13&18&-14&5&8&10&-19&19\\-20&-16&-11&8&5&-19&1&-3\\1&6&-16&-20&-4&19&-3&15\end{bmatrix}\end{align*}$

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Example Question #1 : Symmetric Matrices

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

Possible Answers:

$\begin{bmatrix}-3&-18&-19&32&-8\\-9&5&3&12&4\\-10&-5&12&-20&-10\\25&20&-11&-6&-7\\1&11&-1&2&-27\end{bmatrix}$ $\displaystyle \begin{bmatrix}-3&-18&-19&32&-8\\-9&5&3&12&4\\-10&-5&12&-20&-10\\25&20&-11&-6&-7\\1&11&-1&2&-27\end{bmatrix}$

$\begin{bmatrix}-12&-16&-3&16&17\\-16&14&-7&-7&-4\\-3&-7&-11&7&-11\\16&-7&7&19&7\\17&-4&-11&7&-17\end{bmatrix}$ $\displaystyle \begin{bmatrix}-12&-16&-3&16&17\\-16&14&-7&-7&-4\\-3&-7&-11&7&-11\\16&-7&7&19&7\\17&-4&-11&7&-17\end{bmatrix}$

$\begin{bmatrix}13&6&-9&-6&18\\-15&-5&15&-19&4\\14&-6&9&-19&9\\-15&-5&15&-19&4\\13&6&-9&-6&18\end{bmatrix}$ $\displaystyle \begin{bmatrix}13&6&-9&-6&18\\-15&-5&15&-19&4\\14&-6&9&-19&9\\-15&-5&15&-19&4\\13&6&-9&-6&18\end{bmatrix}$

$\begin{bmatrix}5&-5&13&-5&5\\8&-6&20&-6&8\\15&-6&5&-6&15\\7&19&-6&19&7\\-4&-16&-10&-16&-4\end{bmatrix}$ $\displaystyle \begin{bmatrix}5&-5&13&-5&5\\8&-6&20&-6&8\\15&-6&5&-6&15\\7&19&-6&19&7\\-4&-16&-10&-16&-4\end{bmatrix}$

Correct answer:

Explanation:

$\begin{align*}&\text{A symmetric matrix is one in which all values are mirrored across the diagonal. In other words}\text{it is equal to its transpose:}\\&A=A^{T}\\&\text{Recall that a transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, etc.}\\&\begin{bmatrix}-12&-16&-3&16&17\\-16&14&-7&-7&-4\\-3&-7&-11&7&-11\\16&-7&7&19&7\\17&-4&-11&7&-17\end{bmatrix}\\&\text{Is a symmetric matrix.}\end{align*}$ $\displaystyle \begin{align*}&\text{A symmetric matrix is one in which all values are mirrored across the diagonal. In other words}\text{it is equal to its transpose:}\\&A=A^{T}\\&\text{Recall that a transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, etc.}\\&\begin{bmatrix}-12&-16&-3&16&17\\-16&14&-7&-7&-4\\-3&-7&-11&7&-11\\16&-7&7&19&7\\17&-4&-11&7&-17\end{bmatrix}\\&\text{Is a symmetric matrix.}\end{align*}$

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Example Question #1 : Symmetric Matrices

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

Possible Answers:

$\begin{bmatrix}3&12&23&-4\\4&12&-11&4\\14&-18&-8&-11\\-12&11&-2&-11\end{bmatrix}$ $\displaystyle \begin{bmatrix}3&12&23&-4\\4&12&-11&4\\14&-18&-8&-11\\-12&11&-2&-11\end{bmatrix}$

$\begin{bmatrix}-2&18&4&-12\\18&11&15&12\\4&15&-9&-4\\-12&12&-4&9\end{bmatrix}$ $\displaystyle \begin{bmatrix}-2&18&4&-12\\18&11&15&12\\4&15&-9&-4\\-12&12&-4&9\end{bmatrix}$

$\begin{bmatrix}-18&18&18&-18\\17&-2&-2&17\\9&7&7&9\\7&9&9&7\end{bmatrix}$ $\displaystyle \begin{bmatrix}-18&18&18&-18\\17&-2&-2&17\\9&7&7&9\\7&9&9&7\end{bmatrix}$

$\begin{bmatrix}13&14&-2&-20\\-14&16&7&-9\\-14&16&7&-9\\13&14&-2&-20\end{bmatrix}$ $\displaystyle \begin{bmatrix}13&14&-2&-20\\-14&16&7&-9\\-14&16&7&-9\\13&14&-2&-20\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-2&18&4&-12\\18&11&15&12\\4&15&-9&-4\\-12&12&-4&9\end{bmatrix}$ $\displaystyle \begin{bmatrix}-2&18&4&-12\\18&11&15&12\\4&15&-9&-4\\-12&12&-4&9\end{bmatrix}$

Explanation:

$\begin{align*}&\text{A symmetric matrix is one in which all values are mirrored across the diagonal. In other words}\text{it is equal to its transpose:}\\&A=A^{T}\\&\text{Recall that a transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, etc.}\\&\begin{bmatrix}-2&18&4&-12\\18&11&15&12\\4&15&-9&-4\\-12&12&-4&9\end{bmatrix}\\&\text{Is a symmetric matrix.}\end{align*}$ $\displaystyle \begin{align*}&\text{A symmetric matrix is one in which all values are mirrored across the diagonal. In other words}\text{it is equal to its transpose:}\\&A=A^{T}\\&\text{Recall that a transpose of a matrix is one in which the first row becomes the first column,}\\&\text{the second row the second column, etc.}\\&\begin{bmatrix}-2&18&4&-12\\18&11&15&12\\4&15&-9&-4\\-12&12&-4&9\end{bmatrix}\\&\text{Is a symmetric matrix.}\end{align*}$

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Example Question #6 : Symmetric Matrices

$\begin{align*}&\text{Which of the following matrices is symmetric?}\end{align*}$ $\displaystyle \begin{align*}&\text{Which of the following matrices is symmetric?}\end{align*}$

Possible Answers:

$\begin{bmatrix}-2&5&-10&-10&5&-2\\8&-3&-2&-2&-3&8\\5&-14&7&7&-14&5\\18&-17&-16&-16&-17&18\\-10&-4&-2&-2&-4&-10\\-11&-15&-13&-13&-15&-11\end{bmatrix}$ $\displaystyle \begin{bmatrix}-2&5&-10&-10&5&-2\\8&-3&-2&-2&-3&8\\5&-14&7&7&-14&5\\18&-17&-16&-16&-17&18\\-10&-4&-2&-2&-4&-10\\-11&-15&-13&-13&-15&-11\end{bmatrix}$

$\begin{bmatrix}-10&0&-17&-20&-13&18\\18&1&-11&-16&14&-15\\-16&-15&14&-9&11&-13\\-16&-15&14&-9&11&-13\\18&1&-11&-16&14&-15\\-10&0&-17&-20&-13&18\end{bmatrix}$ $\displaystyle \begin{bmatrix}-10&0&-17&-20&-13&18\\18&1&-11&-16&14&-15\\-16&-15&14&-9&11&-13\\-16&-15&14&-9&11&-13\\18&1&-11&-16&14&-15\\-10&0&-17&-20&-13&18\end{bmatrix}$

$\begin{bmatrix}-7&16&-4&18&4&1\\16&-6&3&9&11&18\\-4&3&-17&3&-18&-3\\18&9&3&0&4&1\\4&11&-18&4&-4&-6\\1&18&-3&1&-6&16\end{bmatrix}$ $\displaystyle \begin{bmatrix}-7&16&-4&18&4&1\\16&-6&3&9&11&18\\-4&3&-17&3&-18&-3\\18&9&3&0&4&1\\4&11&-18&4&-4&-6\\1&18&-3&1&-6&16\end{bmatrix}$

$\begin{bmatrix}-26&11&16&17&-4&-1\\20&-22&-2&9&8&-17\\24&7&-19&4&-19&2\\9&2&-4&8&21&5\\5&-1&-27&14&-14&-1\\6&-8&10&-3&-9&6\end{bmatrix}$ $\displaystyle \begin{bmatrix}-26&11&16&17&-4&-1\\20&-22&-2&9&8&-17\\24&7&-19&4&-19&2\\9&2&-4&8&21&5\\5&-1&-27&14&-14&-1\\6&-8&10&-3&-9&6\end{bmatrix}$

Correct answer:

Explanation:

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

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Example Question #3 : Symmetric Matrices

$\begin{align*}&\text{Which of the following matrices is symmetric?}\end{align*}$ $\displaystyle \begin{align*}&\text{Which of the following matrices is symmetric?}\end{align*}$

Possible Answers:

$\begin{bmatrix}-12&16&13&19&26&0&2\\24&5&-19&29&-19&16&-10\\21&-26&16&13&-17&16&7\\12&20&4&22&18&-3&-12\\17&-26&-9&10&-15&6&11\\-9&23&8&6&-3&8&8\\10&-18&0&-21&19&15&4\end{bmatrix}$ $\displaystyle \begin{bmatrix}-12&16&13&19&26&0&2\\24&5&-19&29&-19&16&-10\\21&-26&16&13&-17&16&7\\12&20&4&22&18&-3&-12\\17&-26&-9&10&-15&6&11\\-9&23&8&6&-3&8&8\\10&-18&0&-21&19&15&4\end{bmatrix}$

$\begin{bmatrix}-3&1&-4&19&8&-5&20\\1&-4&-5&5&-11&0&19\\-4&-5&15&-11&20&-18&14\\19&5&-11&6&3&-9&-4\\8&-11&20&3&1&6&-4\\-5&0&-18&-9&6&2&20\\20&19&14&-4&-4&20&-15\end{bmatrix}$ $\displaystyle \begin{bmatrix}-3&1&-4&19&8&-5&20\\1&-4&-5&5&-11&0&19\\-4&-5&15&-11&20&-18&14\\19&5&-11&6&3&-9&-4\\8&-11&20&3&1&6&-4\\-5&0&-18&-9&6&2&20\\20&19&14&-4&-4&20&-15\end{bmatrix}$

$\begin{bmatrix}18&8&19&-18&16&9&-1\\-13&12&13&17&-19&-13&15\\-15&-14&-19&-5&9&19&-8\\10&1&-9&-10&18&13&13\\-15&-14&-19&-5&9&19&-8\\-13&12&13&17&-19&-13&15\\18&8&19&-18&16&9&-1\end{bmatrix}$ $\displaystyle \begin{bmatrix}18&8&19&-18&16&9&-1\\-13&12&13&17&-19&-13&15\\-15&-14&-19&-5&9&19&-8\\10&1&-9&-10&18&13&13\\-15&-14&-19&-5&9&19&-8\\-13&12&13&17&-19&-13&15\\18&8&19&-18&16&9&-1\end{bmatrix}$

$\begin{bmatrix}4&-16&6&-2&6&-16&4\\0&-11&3&3&3&-11&0\\12&2&-7&13&-7&2&12\\7&-13&-18&-17&-18&-13&7\\-10&-17&15&12&15&-17&-10\\19&0&18&-13&18&0&19\\-10&-20&2&-8&2&-20&-10\end{bmatrix}$ $\displaystyle \begin{bmatrix}4&-16&6&-2&6&-16&4\\0&-11&3&3&3&-11&0\\12&2&-7&13&-7&2&12\\7&-13&-18&-17&-18&-13&7\\-10&-17&15&12&15&-17&-10\\19&0&18&-13&18&0&19\\-10&-20&2&-8&2&-20&-10\end{bmatrix}$

Correct answer:

Explanation:

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

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Example Question #8 : Symmetric Matrices

$\begin{align*}&\text{Which of the following matrices is symmetric?}\end{align*}$ $\displaystyle \begin{align*}&\text{Which of the following matrices is symmetric?}\end{align*}$

Possible Answers:

$\begin{bmatrix}14&19&-17&12&-2&-1&-14\\3&15&13&-4&13&3&-5\\-14&20&-4&8&-20&18&12\\-7&16&5&13&12&-11&-9\\-14&20&-4&8&-20&18&12\\3&15&13&-4&13&3&-5\\14&19&-17&12&-2&-1&-14\end{bmatrix}$ $\displaystyle \begin{bmatrix}14&19&-17&12&-2&-1&-14\\3&15&13&-4&13&3&-5\\-14&20&-4&8&-20&18&12\\-7&16&5&13&12&-11&-9\\-14&20&-4&8&-20&18&12\\3&15&13&-4&13&3&-5\\14&19&-17&12&-2&-1&-14\end{bmatrix}$

$\begin{bmatrix}17&10&13&-18&10&5&3\\10&-14&-7&5&-7&8&-8\\13&-7&20&1&15&-20&-7\\-18&5&1&-15&9&16&-6\\10&-7&15&9&18&-11&3\\5&8&-20&16&-11&-3&11\\3&-8&-7&-6&3&11&20\end{bmatrix}$ $\displaystyle \begin{bmatrix}17&10&13&-18&10&5&3\\10&-14&-7&5&-7&8&-8\\13&-7&20&1&15&-20&-7\\-18&5&1&-15&9&16&-6\\10&-7&15&9&18&-11&3\\5&8&-20&16&-11&-3&11\\3&-8&-7&-6&3&11&20\end{bmatrix}$

$\begin{bmatrix}15&14&-14&0&-14&14&15\\-7&13&-11&-13&-11&13&-7\\10&-12&18&11&18&-12&10\\18&-15&-5&-19&-5&-15&18\\-7&11&5&19&5&11&-7\\-3&8&14&4&14&8&-3\\-11&-7&14&-14&14&-7&-11\end{bmatrix}$ $\displaystyle \begin{bmatrix}15&14&-14&0&-14&14&15\\-7&13&-11&-13&-11&13&-7\\10&-12&18&11&18&-12&10\\18&-15&-5&-19&-5&-15&18\\-7&11&5&19&5&11&-7\\-3&8&14&4&14&8&-3\\-11&-7&14&-14&14&-7&-11\end{bmatrix}$

$\begin{bmatrix}-21&-16&14&-5&-2&17&-20\\-23&-1&-20&-19&25&-25&-2\\22&-13&-3&-12&-19&2&9\\2&-26&-5&-12&10&-26&-16\\-10&17&-11&2&5&30&-11\\25&-16&-5&-18&21&18&2\\-27&5&17&-9&-3&10&1\end{bmatrix}$ $\displaystyle \begin{bmatrix}-21&-16&14&-5&-2&17&-20\\-23&-1&-20&-19&25&-25&-2\\22&-13&-3&-12&-19&2&9\\2&-26&-5&-12&10&-26&-16\\-10&17&-11&2&5&30&-11\\25&-16&-5&-18&21&18&2\\-27&5&17&-9&-3&10&1\end{bmatrix}$

Correct answer:

Explanation:

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

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Example Question #151 : Linear Algebra

$\begin{align*}&\text{Which of the following matrices is symmetric?}\end{align*}$ $\displaystyle \begin{align*}&\text{Which of the following matrices is symmetric?}\end{align*}$

Possible Answers:

$\begin{bmatrix}5&19&8&19&5\\13&-9&-8&-9&13\\3&-15&17&-15&3\\16&-16&-7&-16&16\\12&-20&3&-20&12\end{bmatrix}$ $\displaystyle \begin{bmatrix}5&19&8&19&5\\13&-9&-8&-9&13\\3&-15&17&-15&3\\16&-16&-7&-16&16\\12&-20&3&-20&12\end{bmatrix}$

$\begin{bmatrix}3&-19&-10&-18&-16\\9&16&-19&-8&-7\\-9&-2&17&8&-15\\9&16&-19&-8&-7\\3&-19&-10&-18&-16\end{bmatrix}$ $\displaystyle \begin{bmatrix}3&-19&-10&-18&-16\\9&16&-19&-8&-7\\-9&-2&17&8&-15\\9&16&-19&-8&-7\\3&-19&-10&-18&-16\end{bmatrix}$

$\begin{bmatrix}-13&6&13&2&-17\\6&19&17&14&-7\\13&17&7&-10&-13\\2&14&-10&-6&-15\\-17&-7&-13&-15&4\end{bmatrix}$ $\displaystyle \begin{bmatrix}-13&6&13&2&-17\\6&19&17&14&-7\\13&17&7&-10&-13\\2&14&-10&-6&-15\\-17&-7&-13&-15&4\end{bmatrix}$

$\begin{bmatrix}10&-34&2&14&-4\\-25&19&2&5&-13\\-5&-6&1&-15&-35\\5&-4&-7&24&1\\3&-6&-26&9&-2\end{bmatrix}$ $\displaystyle \begin{bmatrix}10&-34&2&14&-4\\-25&19&2&5&-13\\-5&-6&1&-15&-35\\5&-4&-7&24&1\\3&-6&-26&9&-2\end{bmatrix}$

Correct answer:

Explanation:

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

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Linear Algebra : Symmetric Matrices

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