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Linear Algebra : Linear Algebra

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Example Questions

Example Question #7 : Symmetric Matrices

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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Example Question #8 : Symmetric Matrices

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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Example Question #11 : Symmetric Matrices

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

Possible Answers:

$\begin{bmatrix}-18&19&-4&-33&-16&-35\\28&5&-5&-12&7&14\\-11&-12&14&-15&11&14\\-26&-21&-6&0&14&-1\\-8&-1&2&5&4&-6\\-28&21&23&-10&2&-18\end{bmatrix}$

$\begin{bmatrix}14&6&-12&-3&-4&4\\6&17&17&-13&1&-4\\-12&17&-13&-20&4&-17\\-3&-13&-20&-2&16&9\\-4&1&4&16&-10&-13\\4&-4&-17&9&-13&-13\end{bmatrix}$

$\begin{bmatrix}-14&-8&8&8&-8&-14\\-8&0&4&4&0&-8\\-7&15&1&1&15&-7\\15&-8&-7&-7&-8&15\\3&8&18&18&8&3\\1&-18&-20&-20&-18&1\end{bmatrix}$

$\begin{bmatrix}-12&-18&15&-14&-3&9\\-19&7&16&-10&-3&15\\13&-19&-5&-14&-10&-15\\13&-19&-5&-14&-10&-15\\-19&7&16&-10&-3&15\\-12&-18&15&-14&-3&9\end{bmatrix}$

Correct answer:

$\begin{bmatrix}14&6&-12&-3&-4&4\\6&17&17&-13&1&-4\\-12&17&-13&-20&4&-17\\-3&-13&-20&-2&16&9\\-4&1&4&16&-10&-13\\4&-4&-17&9&-13&-13\end{bmatrix}$

Explanation:

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

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Example Question #12 : Symmetric Matrices

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

Possible Answers:

$\begin{bmatrix}10&5&19&19&5&10\\-19&13&-10&-10&13&-19\\-18&8&17&17&8&-18\\-15&16&9&9&16&-15\\-11&12&4&4&12&-11\\15&-12&-14&-14&-12&15\end{bmatrix}$

$\begin{bmatrix}-18&-17&9&34&33&11\\-10&-15&-12&17&-11&-29\\1&-4&22&-36&-21&9\\26&10&-28&2&32&5\\25&-2&-12&24&-9&30\\3&-20&16&-3&23&10\end{bmatrix}$

$\begin{bmatrix}-8&-17&3&15&-12&4\\-17&-9&6&-4&5&2\\3&6&13&6&-19&0\\15&-4&6&14&-8&-17\\-12&5&-19&-8&-6&-6\\4&2&0&-17&-6&9\end{bmatrix}$

$\begin{bmatrix}14&-1&2&15&10&-13\\-12&-5&-3&7&-13&11\\-2&-4&-7&17&8&-1\\-2&-4&-7&17&8&-1\\-12&-5&-3&7&-13&11\\14&-1&2&15&10&-13\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-8&-17&3&15&-12&4\\-17&-9&6&-4&5&2\\3&6&13&6&-19&0\\15&-4&6&14&-8&-17\\-12&5&-19&-8&-6&-6\\4&2&0&-17&-6&9\end{bmatrix}$

Explanation:

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

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Example Question #13 : Symmetric Matrices

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

Possible Answers:

$\begin{bmatrix}20&-8&-5&-5&-8&20\\19&15&-15&-15&15&19\\19&12&-17&-17&12&19\\-1&-7&-4&-4&-7&-1\\6&9&-17&-17&9&6\\3&-16&17&17&-16&3\end{bmatrix}$

$\begin{bmatrix}-5&1&15&19&12&14\\-17&-12&20&3&2&-12\\12&5&-9&18&10&18\\12&5&-9&18&10&18\\-17&-12&20&3&2&-12\\-5&1&15&19&12&14\end{bmatrix}$

$\begin{bmatrix}-15&-8&24&14&0&14\\-15&6&8&-2&25&-1\\16&1&-21&16&-7&18\\23&-11&7&-26&0&-15\\-7&16&1&9&7&-4\\22&-9&9&-7&-13&12\end{bmatrix}$

$\begin{bmatrix}-2&-6&-14&8&-7&-5\\-6&20&-14&9&0&13\\-14&-14&-15&-14&7&-1\\8&9&-14&13&3&-11\\-7&0&7&3&-18&5\\-5&13&-1&-11&5&-6\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-2&-6&-14&8&-7&-5\\-6&20&-14&9&0&13\\-14&-14&-15&-14&7&-1\\8&9&-14&13&3&-11\\-7&0&7&3&-18&5\\-5&13&-1&-11&5&-6\end{bmatrix}$

Explanation:

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

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Example Question #12 : Symmetric Matrices

$A = \begin{bmatrix} 1& -3 &4 \\ 3& 1& 5\\ -4&-5 & 1\end{bmatrix}$

True or false: is an example of a skew-symmetric matrix.

Possible Answers:

True

False

Correct answer:

False

Explanation:

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

$A^{T} = -A$ .

Interchanging rows and columns, we see that if

$A = \begin{bmatrix} 1& -3 &4 \\ 3& 1& 5\\ -4&-5 & 1\end{bmatrix}$ ,

then

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

can be determined by changing each element in to its additive inverse:

$A = \begin{bmatrix} -1& 3 &-4 \\- 3&- 1& -5\\ 4& 5 &- 1\end{bmatrix}$

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

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Example Question #71 : Operations And Properties

$A = \begin{bmatrix} 0& -3 &4 \\ 3& 0& 5\\ -4&-5 & 0 \end{bmatrix}$

True or false: is an example of a skew-symmetric matrix.

Possible Answers:

True

False

Correct answer:

True

Explanation:

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

$A^{T} = -A$ .

Interchanging rows and columns, we see that if

$A = \begin{bmatrix} 0& -3 &4 \\ 3& 0& 5\\ -4&-5 & 0 \end{bmatrix}$ ,

then

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

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

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Example Question #153 : Linear Algebra

is a three-by-three nonsingular skew-symmetric matrix

Then which of the following must be equal to $A^{T}A^{-1}$ ?

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

$A^{T} = -A$ .

Therefore, by substitution,

$A^{T}A^{-1} = (-A)A^{-1}$

$A^{T}A^{-1} = (-1 A)A^{-1}$

$A^{T}A^{-1} = -1(AA^{-1})$

$A^{T}A^{-1} = -1I$

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

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Example Question #11 : Symmetric Matrices

$A = \begin{bmatrix} a & 2& -3\\ -2&a & 5 \\ 3& -5& a \end{bmatrix}$

Which of the following must be true of for to be a skew-symmetric matrix?

Possible Answers:

Either a = -i or a = i

It is impossible for to be a skew-symmetric matrix regardless of the value of .

a = 0

is a skew-symmetric matrix regardless of the value of .

Either a = -1 or a = 1

Correct answer:

a = 0

Explanation:

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

$A^{T} = -A$ .

Interchanging rows with columns in , we see that if

$A = \begin{bmatrix} a & 2& -3\\ -2&a & 5 \\ 3& -5& a \end{bmatrix}$

then

$A ^{T}= \begin{bmatrix} a &- 2& 3\\ 2&a &- 5 \\ - 3& 5& a \end{bmatrix}$

Also, by changing each entry in to its additive inverse, we see that

$-A = \begin{bmatrix} -a & -2& 3\\ 2&-a & -5 \\- 3& 5& -a \end{bmatrix}$

Setting the two equal to each other, we see that:

$\begin{bmatrix} a &- 2& 3\\ 2&a &- 5 \\ - 3& 5& a \end{bmatrix}= \begin{bmatrix} -a & -2& 3\\ 2&-a & -5 \\- 3& 5& -a \end{bmatrix}$

The non-diagonal elements - all constants - are all equal. Looking at the diagonal elements, we see that it is necessary and sufficient for a = -a ; that is, must be its own additive inverse. The only such number is 0, so a= 0 .

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Example Question #11 : Symmetric Matrices

is a square matrix.

Which must be true of $A - A ^{T}$ ?

Possible Answers:

$A - A ^{T}$ must be symmetric.

Neither of the other statements is correct.

$A - A ^{T}$ must be skew-symmetric.

Correct answer:

$A - A ^{T}$ must be skew-symmetric.

Explanation:

Let be a three-by-three matrix - this reasoning extends to matrices of any size.

Let

$A = \begin{bmatrix} a& b & c \\ d & e & f \\ g & h & j \end{bmatrix}$

$A^{T}$ is the transpose of the matrix, which is formed when its rows are interchanged with its columns; this is

$A^{T} = \begin{bmatrix} a& d & g \\ b&e & h\\ c & f & j \end{bmatrix}$

Subtract elementwise:

$A - A ^{T}$

$= \begin{bmatrix} a& b & c \\ d & e & f \\ g & h & j \end{bmatrix} - \begin{bmatrix} a& d & g \\ b&e & h\\ c & f & j \end{bmatrix}$

$= \begin{bmatrix} a-a & b-d & c-g \\ d-b & e -e& f -h\\ g-c & h-f & j-j \end{bmatrix}$

$= \begin{bmatrix} 0 & b-d & c-g \\ -b+d & 0 & f -h\\ -c+g & -f +h& 0 \end{bmatrix}$

$= \begin{bmatrix} 0 & b-d & c-g \\ -(b-d )& 0 & f -h\\ - (c-g ) & -(f -h)& 0 \end{bmatrix}$

A matrix is symmetric if and only it is equal to its transpose; it is skew-symmetric if and only if it is equal to the additive inverse of its transpose. Interchanging rows and columns in $A - A ^{T}$ , we see that

$(A - A ^{T})^{T}= \begin{bmatrix} 0 & -(b-d ) & -(c-g) \\ b-d& 0 & -(f -h)\\ c-g & f -h& 0 \end{bmatrix}$ .

Each element in $(A - A ^{T})^{T}$ is the additive inverse of the corresponding element in $A - A ^{T}$ , so

$(A - A ^{T})^{T} = - (A - A ^{T})$ ,

making $A - A ^{T}$ a skew-symmetric matrix.

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Linear Algebra : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #7 : Symmetric Matrices

Example Question #8 : Symmetric Matrices

Example Question #11 : Symmetric Matrices

Example Question #12 : Symmetric Matrices

Example Question #13 : Symmetric Matrices

Example Question #12 : Symmetric Matrices

Example Question #71 : Operations And Properties

Example Question #153 : Linear Algebra

Example Question #11 : Symmetric Matrices

Example Question #11 : Symmetric Matrices

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