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Linear Algebra : Operations and Properties

Study concepts, example questions & explanations for Linear Algebra

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Example Questions

Example Question #1 : The Transpose

Which is the transpose of $\begin{bmatrix} 1 &8 & 4\\2 &-3 &0 \end{bmatrix}$ ?

Possible Answers:

$\begin{bmatrix} 1 & 2& 8\\ 4& -3& 0\end{bmatrix}$

$\begin{bmatrix} 2 & 1\\ -3& 8\\ 0& 4\end{bmatrix}$

$\begin{bmatrix} 1 &2 \\8 & -3\\ 4& 0\end{bmatrix}$

$\begin{bmatrix} 1 &4 \\2 &-3 \\8 &0 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 1 &2 \\8 & -3\\ 4& 0\end{bmatrix}$

Explanation:

The transverse is the matrix where the columns are now the corresponding rows - the first column is now the first row, the second column is now the second row, etc.

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Example Question #2 : The Transpose

Find the transpose of matrix A.

$A=\begin{bmatrix} 7&4 \\ 2&8 \\ 1&3 \end{bmatrix}$

Possible Answers:

$A^{T}= \begin{bmatrix} 7&4 \\ 2& 8\\ 1&3 \end{bmatrix}$

$A^{T}=\begin{bmatrix} 7& 2&1 \\ 4& 8&3 \end{bmatrix}$

$A^{T}=\begin{bmatrix} 4&8 &3 \\ 7&2 &1 \end{bmatrix}$

$A^{T}=\begin{bmatrix} 4&7 \\ 8& 2\\ 3& 1\end{bmatrix}$

None of the other answers

Correct answer:

$A^{T}=\begin{bmatrix} 7& 2&1 \\ 4& 8&3 \end{bmatrix}$

Explanation:

For a 3x2 matrix A, the transpose of A is a 2x3 matrix, where the columns are formed from the corresponding rows of A.

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Example Question #1 : The Transpose

Find the transpose of matrix A.

$A=\begin{bmatrix} 2& 4&6 &8 \\1 & 3& 5&7 \end{bmatrix}$

Possible Answers:

$A^{T}=\begin{bmatrix} 1&5 \\2 & 6\\ 3&7 \\ 4&8 \end{bmatrix}$

$A^{T}=\begin{bmatrix} 2&1 \\ 4&3 \\6 &5 \\ 8&7 \end{bmatrix}$

None of the other answers

$A^{T}=\begin{bmatrix} 1& 2\\ 3& 4\\ 5& 6\\7 & 8\end{bmatrix}$

$A^{T}=\begin{bmatrix} 1& 3&5 &7 \\ 2& 4&6 &8 \end{bmatrix}$

Correct answer:

$A^{T}=\begin{bmatrix} 2&1 \\ 4&3 \\6 &5 \\ 8&7 \end{bmatrix}$

Explanation:

For a 2x4 matrix A, the transpose of A is a 4x2 matrix, where the columns are formed from the corresponding rows of A.

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Example Question #11 : The Transpose

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

Possible Answers:

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

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

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

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

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 #111 : Linear Algebra

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

Possible Answers:

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

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

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

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

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 #12 : The Transpose

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

Possible Answers:

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

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

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

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

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 #31 : 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:

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

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

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

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

Correct answer:

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

Explanation:

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Example Question #14 : The Transpose

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

Possible Answers:

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

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

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

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

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 #15 : The Transpose

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

Possible Answers:

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

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

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

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

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 #16 : The Transpose

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

Possible Answers:

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

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

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

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

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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Linear Algebra : Operations and Properties

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #1 : The Transpose

Example Question #2 : The Transpose

Example Question #1 : The Transpose

Example Question #11 : The Transpose

Example Question #111 : Linear Algebra

Example Question #12 : The Transpose

Example Question #31 : Operations And Properties

Example Question #14 : The Transpose

Example Question #15 : The Transpose

Example Question #16 : The Transpose

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