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Linear Algebra : The Transpose

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Example Questions

Example Question #31 : Operations And Properties

$\begin{align*}&\text{Find the transpose of }A=\begin{bmatrix}-15&3\\-2&18\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}-2&-15\\18&3\end{bmatrix}$

$\begin{bmatrix}-15&-2\\3&18\end{bmatrix}$

$\begin{bmatrix}3&18\\-15&-2\end{bmatrix}$

$\begin{bmatrix}18&3\\-2&-15\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-15&-2\\3&18\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Determining the transpose of a matrix is relatively simple, though attention to detail should be exercised.}\\&\text{In creating the transpose, the first row becomes the first column, the second row the second column, etc:}\\&A=\begin{bmatrix}-15&3\\-2&18\end{bmatrix}\\&A^{T}=\begin{bmatrix}-15&-2\\3&18\end{bmatrix}\end{align*}$

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Example Question #32 : Operations And Properties

$\begin{align*}&\text{Find the transpose of }\begin{bmatrix}1&2\\-2&-6\\-8&-20\\-18&-14\\16&15\\20&3\end{bmatrix}^{T}\end{align*}$

Possible Answers:

$\begin{bmatrix}1&-2&-8&-18&16&20\\2&-6&-20&-14&15&3\end{bmatrix}$

$\begin{bmatrix}20&16&-18&-8&-2&1\\3&15&-14&-20&-6&2\end{bmatrix}$

$\begin{bmatrix}3&15&-14&-20&-6&2\\20&16&-18&-8&-2&1\end{bmatrix}$

$\begin{bmatrix}1&2\\-2&-6\\-8&-20\\-18&-14\\16&15\\20&3\end{bmatrix}$

Correct answer:

$\begin{bmatrix}1&2\\-2&-6\\-8&-20\\-18&-14\\16&15\\20&3\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Note the superscript, T, in our problem statement; this denotes the transpose.}\\&\text{The transpose of the transpose of a matrix is the orignal matrix, so:}\\&\begin{bmatrix}1&2\\-2&-6\\-8&-20\\-18&-14\\16&15\\20&3\end{bmatrix}^{TT}=\begin{bmatrix}1&2\\-2&-6\\-8&-20\\-18&-14\\16&15\\20&3\end{bmatrix}\end{align*}$

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Example Question #33 : Operations And Properties

$\begin{align*}&\text{Find }\begin{bmatrix}1&-1&0\\3&13&-10\end{bmatrix}^{T}\end{align*}$

Possible Answers:

$\begin{bmatrix}-10&0\\13&-1\\3&1\end{bmatrix}$

$\begin{bmatrix}0&-10\\-1&13\\1&3\end{bmatrix}$

$\begin{bmatrix}1&3\\-1&13\\0&-10\end{bmatrix}$

$\begin{bmatrix}3&1\\13&-1\\-10&0\end{bmatrix}$

Correct answer:

$\begin{bmatrix}1&3\\-1&13\\0&-10\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Superscript T denotes the transpose of a matrix. Finding this transpose requies only}\\&\text{treating the first row as the first column, the second as row the second column, etc:}\\&\begin{bmatrix}1&-1&0\\3&13&-10\end{bmatrix}^{T}=\begin{bmatrix}1&3\\-1&13\\0&-10\end{bmatrix}\end{align*}$

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Example Question #34 : Operations And Properties

$\begin{align*}&\text{Find the transpose of }A=\begin{bmatrix}-13&-12&15&1&-18\\0&3&-17&-1&-13\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}0&-13\\3&-12\\-17&15\\-1&1\\-13&-18\end{bmatrix}$

$\begin{bmatrix}-13&0\\-12&3\\15&-17\\1&-1\\-18&-13\end{bmatrix}$

$\begin{bmatrix}-13&-18\\-1&1\\-17&15\\3&-12\\0&-13\end{bmatrix}$

$\begin{bmatrix}-18&-13\\1&-1\\15&-17\\-12&3\\-13&0\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-13&0\\-12&3\\15&-17\\1&-1\\-18&-13\end{bmatrix}$

Explanation:

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Example Question #35 : Operations And Properties

$\begin{align*}&\text{Find the transpose of }A=\begin{bmatrix}10&20&2&-7&13&-3&11&-8\\-6&13&-13&0&3&6&4&-13\\-19&5&-14&-10&-12&-16&12&-12\\-12&-18&-13&-18&-17&-15&8&-2\\-2&-14&0&13&-19&-6&-15&-2\\-17&16&1&-16&-15&-9&-14&-4\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}-17&-2&-12&-19&-6&10\\16&-14&-18&5&13&20\\1&0&-13&-14&-13&2\\-16&13&-18&-10&0&-7\\-15&-19&-17&-12&3&13\\-9&-6&-15&-16&6&-3\\-14&-15&8&12&4&11\\-4&-2&-2&-12&-13&-8\end{bmatrix}$

$\begin{bmatrix}10&-6&-19&-12&-2&-17\\20&13&5&-18&-14&16\\2&-13&-14&-13&0&1\\-7&0&-10&-18&13&-16\\13&3&-12&-17&-19&-15\\-3&6&-16&-15&-6&-9\\11&4&12&8&-15&-14\\-8&-13&-12&-2&-2&-4\end{bmatrix}$

$\begin{bmatrix}-8&-13&-12&-2&-2&-4\\11&4&12&8&-15&-14\\-3&6&-16&-15&-6&-9\\13&3&-12&-17&-19&-15\\-7&0&-10&-18&13&-16\\2&-13&-14&-13&0&1\\20&13&5&-18&-14&16\\10&-6&-19&-12&-2&-17\end{bmatrix}$

$\begin{bmatrix}-4&-2&-2&-12&-13&-8\\-14&-15&8&12&4&11\\-9&-6&-15&-16&6&-3\\-15&-19&-17&-12&3&13\\-16&13&-18&-10&0&-7\\1&0&-13&-14&-13&2\\16&-14&-18&5&13&20\\-17&-2&-12&-19&-6&10\end{bmatrix}$

Correct answer:

$\begin{bmatrix}10&-6&-19&-12&-2&-17\\20&13&5&-18&-14&16\\2&-13&-14&-13&0&1\\-7&0&-10&-18&13&-16\\13&3&-12&-17&-19&-15\\-3&6&-16&-15&-6&-9\\11&4&12&8&-15&-14\\-8&-13&-12&-2&-2&-4\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Determining the transpose of a matrix is relatively simple,}\\&\text{though attention to detail should be exercised.}\\&\text{In creating the transpose, the first row becomes the first column,}\\&\text{the second row the second column, etc:}\\&A=\begin{bmatrix}10&20&2&-7&13&-3&11&-8\\-6&13&-13&0&3&6&4&-13\\-19&5&-14&-10&-12&-16&12&-12\\-12&-18&-13&-18&-17&-15&8&-2\\-2&-14&0&13&-19&-6&-15&-2\\-17&16&1&-16&-15&-9&-14&-4\end{bmatrix}\\&A^{T}=\begin{bmatrix}10&-6&-19&-12&-2&-17\\20&13&5&-18&-14&16\\2&-13&-14&-13&0&1\\-7&0&-10&-18&13&-16\\13&3&-12&-17&-19&-15\\-3&6&-16&-15&-6&-9\\11&4&12&8&-15&-14\\-8&-13&-12&-2&-2&-4\end{bmatrix}\end{align*}$

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Example Question #36 : Operations And Properties

$\begin{align*}&\text{Find }A^{T}\text{ for }A=\begin{bmatrix}5&-16&2\\-4&-15&20\\8&-1&11\\8&-11&-15\\17&20&-9\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}2&20&11&-15&-9\\-16&-15&-1&-11&20\\5&-4&8&8&17\end{bmatrix}$

$\begin{bmatrix}17&8&8&-4&5\\20&-11&-1&-15&-16\\-9&-15&11&20&2\end{bmatrix}$

$\begin{bmatrix}-9&-15&11&20&2\\20&-11&-1&-15&-16\\17&8&8&-4&5\end{bmatrix}$

$\begin{bmatrix}5&-4&8&8&17\\-16&-15&-1&-11&20\\2&20&11&-15&-9\end{bmatrix}$

Correct answer:

$\begin{bmatrix}5&-4&8&8&17\\-16&-15&-1&-11&20\\2&20&11&-15&-9\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Finding the transpose of a matrix is simply a matter of making}\\&\text{the first row the first column, the second row the second column, etc:}\\&A=\begin{bmatrix}5&-16&2\\-4&-15&20\\8&-1&11\\8&-11&-15\\17&20&-9\end{bmatrix}\\&A^{T}=\begin{bmatrix}5&-4&8&8&17\\-16&-15&-1&-11&20\\2&20&11&-15&-9\end{bmatrix}\end{align*}$

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Example Question #37 : Operations And Properties

$\begin{align*}&\text{Find the transpose of }\begin{bmatrix}6&-14\\14&8\\15&-15\\18&-16\\-5&20\\-7&13\end{bmatrix}^{T}\end{align*}$

Possible Answers:

$\begin{bmatrix}-7&-5&18&15&14&6\\13&20&-16&-15&8&-14\end{bmatrix}$

$\begin{bmatrix}6&-14\\14&8\\15&-15\\18&-16\\-5&20\\-7&13\end{bmatrix}$

$\begin{bmatrix}13&20&-16&-15&8&-14\\-7&-5&18&15&14&6\end{bmatrix}$

$\begin{bmatrix}6&14&15&18&-5&-7\\-14&8&-15&-16&20&13\end{bmatrix}$

Correct answer:

$\begin{bmatrix}6&-14\\14&8\\15&-15\\18&-16\\-5&20\\-7&13\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Note the superscript, T, in our problem statement; this denotes the transpose.}\\&\text{The transpose of the transpose of a matrix is the orignal matrix, so:}\\&\begin{bmatrix}6&-14\\14&8\\15&-15\\18&-16\\-5&20\\-7&13\end{bmatrix}^{TT}=\begin{bmatrix}6&-14\\14&8\\15&-15\\18&-16\\-5&20\\-7&13\end{bmatrix}\end{align*}$

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Example Question #41 : Operations And Properties

$\begin{align*}&\text{Find the transpose of }\begin{bmatrix}-4&-4&13&-8&13\\-6&-17&18&12&3\\1&-9&-6&-7&18\\18&-17&-18&-7&6\end{bmatrix}^{T}\end{align*}$

Possible Answers:

$\begin{bmatrix}18&1&-6&-4\\-17&-9&-17&-4\\-18&-6&18&13\\-7&-7&12&-8\\6&18&3&13\end{bmatrix}$

$\begin{bmatrix}6&18&3&13\\-7&-7&12&-8\\-18&-6&18&13\\-17&-9&-17&-4\\18&1&-6&-4\end{bmatrix}$

$\begin{bmatrix}-4&-6&1&18\\-4&-17&-9&-17\\13&18&-6&-18\\-8&12&-7&-7\\13&3&18&6\end{bmatrix}$

$\begin{bmatrix}-4&-4&13&-8&13\\-6&-17&18&12&3\\1&-9&-6&-7&18\\18&-17&-18&-7&6\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-4&-4&13&-8&13\\-6&-17&18&12&3\\1&-9&-6&-7&18\\18&-17&-18&-7&6\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Note the superscript, T, in our problem statement; this denotes the transpose.}\\&\text{The transpose of the transpose of a matrix is the orignal matrix, so:}\\&\begin{bmatrix}-4&-4&13&-8&13\\-6&-17&18&12&3\\1&-9&-6&-7&18\\18&-17&-18&-7&6\end{bmatrix}^{TT}=\begin{bmatrix}-4&-4&13&-8&13\\-6&-17&18&12&3\\1&-9&-6&-7&18\\18&-17&-18&-7&6\end{bmatrix}\end{align*}$

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Example Question #42 : Operations And Properties

$\begin{align*}&\text{Find }A^{T}\text{ for }A=\begin{bmatrix}-11&14\\-18&7\\7&-6\\-17&-9\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}-17&7&-18&-11\\-9&-6&7&14\end{bmatrix}$

$\begin{bmatrix}14&7&-6&-9\\-11&-18&7&-17\end{bmatrix}$

$\begin{bmatrix}-11&-18&7&-17\\14&7&-6&-9\end{bmatrix}$

$\begin{bmatrix}-9&-6&7&14\\-17&7&-18&-11\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-11&-18&7&-17\\14&7&-6&-9\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Finding the transpose of a matrix is simply a matter of making}\\&\text{the first row the first column, the second row the second column, etc:}\\&A=\begin{bmatrix}-11&14\\-18&7\\7&-6\\-17&-9\end{bmatrix}\\&A^{T}=\begin{bmatrix}-11&-18&7&-17\\14&7&-6&-9\end{bmatrix}\end{align*}$

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Example Question #121 : Linear Algebra

$\begin{align*}&\text{Find the transpose of }A=\begin{bmatrix}14&-3&18&17&-12\\3&2&-16&20&-11\\-8&-14&-15&17&16\\15&-10&3&14&-8\\-17&-7&15&4&-3\\8&-10&17&12&20\\-14&14&-7&-9&-10\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}14&3&-8&15&-17&8&-14\\-3&2&-14&-10&-7&-10&14\\18&-16&-15&3&15&17&-7\\17&20&17&14&4&12&-9\\-12&-11&16&-8&-3&20&-10\end{bmatrix}$

$\begin{bmatrix}-14&8&-17&15&-8&3&14\\14&-10&-7&-10&-14&2&-3\\-7&17&15&3&-15&-16&18\\-9&12&4&14&17&20&17\\-10&20&-3&-8&16&-11&-12\end{bmatrix}$

$\begin{bmatrix}-12&-11&16&-8&-3&20&-10\\17&20&17&14&4&12&-9\\18&-16&-15&3&15&17&-7\\-3&2&-14&-10&-7&-10&14\\14&3&-8&15&-17&8&-14\end{bmatrix}$

$\begin{bmatrix}-10&20&-3&-8&16&-11&-12\\-9&12&4&14&17&20&17\\-7&17&15&3&-15&-16&18\\14&-10&-7&-10&-14&2&-3\\-14&8&-17&15&-8&3&14\end{bmatrix}$

Correct answer:

$\begin{bmatrix}14&3&-8&15&-17&8&-14\\-3&2&-14&-10&-7&-10&14\\18&-16&-15&3&15&17&-7\\17&20&17&14&4&12&-9\\-12&-11&16&-8&-3&20&-10\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Determining the transpose of a matrix is relatively simple,}\\&\text{though attention to detail should be exercised.}\\&\text{In creating the transpose, the first row becomes the first column,}\\&\text{the second row the second column, etc:}\\&A=\begin{bmatrix}14&-3&18&17&-12\\3&2&-16&20&-11\\-8&-14&-15&17&16\\15&-10&3&14&-8\\-17&-7&15&4&-3\\8&-10&17&12&20\\-14&14&-7&-9&-10\end{bmatrix}\\&A^{T}=\begin{bmatrix}14&3&-8&15&-17&8&-14\\-3&2&-14&-10&-7&-10&14\\18&-16&-15&3&15&17&-7\\17&20&17&14&4&12&-9\\-12&-11&16&-8&-3&20&-10\end{bmatrix}\end{align*}$

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Linear Algebra : The Transpose

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #31 : Operations And Properties

Example Question #32 : Operations And Properties

Example Question #33 : Operations And Properties

Example Question #34 : Operations And Properties

Example Question #35 : Operations And Properties

Example Question #36 : Operations And Properties

Example Question #37 : Operations And Properties

Example Question #41 : Operations And Properties

Example Question #42 : Operations And Properties

Example Question #121 : Linear Algebra

All Linear Algebra Resources

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