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Linear Algebra : Operations and Properties

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Example Questions

Example Question #38 : The Transpose

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

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

$A^{T}$ is the transpose of - the result of interchanging the rows of with its columns. $A^{*}$ is the conjugate transpose of - the result of changing each entry of $A^{T}$ to its complex conjugate. Therefore, if

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

we can find $A^{T}$ by simply changing each entry in $A^{*}$ to its complex conjugate:

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

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

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Example Question #32 : The Transpose

$A^{*} = \begin{bmatrix} 3+ 5i & 0 & 0& 0 \\ 2 - 5i & 3+ 6i & 0 & 0 \\1 + 6i &2 - 8 i & 4 + 8 i & 0 \\ 1-2i & 3 - 7i & 2 - 7i & 3- 5i \end{bmatrix}$

True or false: is an upper triangular matrix.

Possible Answers:

False

True

Correct answer:

True

Explanation:

$A^{*}$ is the result of interchanging rows of with columns, then changing each entry to its complex conjugate. Also, is equal to $(A^{*} )^{*} = A$ , so perform the same process on $A^{*}$ :

$A^{*} = \begin{bmatrix} 3+ 5i & 0 & 0& 0 \\ 2 - 5i & 3+ 6i & 0 & 0 \\1 + 6i &2 - 8 i & 4 + 8 i & 0 \\ 1-2i & 3 - 7i & 2 - 7i & 3- 5i \end{bmatrix}$

$(A^{*})^{T} = \begin{bmatrix} 3+ 5i & 2 - 5i & 1 + 6i & 1-2i \\ 0& 3+ 6i & 2 - 8 i & 3 - 7i \\ 0& 0& 4 + 8 i & 2 - 7i \\ 0& 0 & 0 & 3- 5i \end{bmatrix}$

$A= (A^{*})^{*} = \begin{bmatrix} 3- 5i & 2 + 5i & 1 - 6i & 1+2i \\ {\color{Red} \textbf{0} }& 3-6i & 2 + 8 i & 3+ 7i \\ {\color{Red} \textbf{0} } & {\color{Red} \textbf{0} } & 4 - 8 i & 2 +7i \\ {\color{Red} \textbf{0} }& {\color{Red} \textbf{0} } & {\color{Red} \textbf{0} } & 3+ 5i \end{bmatrix}$

A matrix is upper triangular if all elements below its main (upper-left corner to lower right corner) diagonal are equal to 0. These elements in are displayed in red above. Since all of the lower-triangular elements of are zeroes, is upper triangular.

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Example Question #32 : The Transpose

$A = \begin{bmatrix} -3 & 1 & 4 \\ 2 & -1 & 3 \end{bmatrix}$

Determine $A+A^{T}$ .

Possible Answers:

$\begin{bmatrix} -6 & 3 & 4 \\ 3 & -2& 3 \\ 4 & 3 & 0 \end{bmatrix}$

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

$A+A^{T}$ is undefined.

$\begin{bmatrix} -6 & 3 \\ 3 & -2 \end{bmatrix}$

Correct answer:

$A+A^{T}$ is undefined.

Explanation:

is a two-by-three matrix. It follows that its transpose, $A^{T}$ , the result of switching rows with columns, is a three-by-two matrix. Since and $A^{T}$ have different dimensions, $A+A^{T}$ is an undefined expression.

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Example Question #41 : The Transpose

True or false; The set of all $n \times n$ symmetric matrices is a subspace of all $n \times n$ matrices.

Possible Answers:

True

False

Correct answer:

True

Explanation:

Without being too abstract, it is easy to convince oneself that this is true. We have to check the 3 criteria for a subspace.

1. Closure under vector addition

Adding together two symmetric matrices will always result in another symmetric matrix.

2. Closure under scalar multiplication.

Multiplying a symmetric matrix by a scalar will also always give you another symmetric matrix

3. The zero vector (matrix in this case) is also in the subset

Indeed the zero vector itself is a symmetric matrix.

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Example Question #1 : Symmetric Matrices

Which matrix is symmetric?

Possible Answers:

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

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

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

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

Correct answer:

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

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

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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Example Question #3 : Symmetric Matrices

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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Example Question #2 : Symmetric Matrices

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

Possible Answers:

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

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

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

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

Correct answer:

Explanation:

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

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Example Question #63 : Operations And Properties

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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Example Question #4 : Symmetric Matrices

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

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

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

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Linear Algebra : Operations and Properties

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #38 : The Transpose

Example Question #32 : The Transpose

Example Question #32 : The Transpose

Example Question #41 : The Transpose

Example Question #1 : Symmetric Matrices

Example Question #2 : Symmetric Matrices

Example Question #3 : Symmetric Matrices

Example Question #2 : Symmetric Matrices

Example Question #63 : Operations And Properties

Example Question #4 : Symmetric Matrices

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