The Transpose - Linear Algebra

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Question

Find the transpose of Matrix .

$A=\begin{bmatrix} 1& 4\ 5& 8 \end{bmatrix}$

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Answer

To find the transpose, we need to make columns into rows.

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Question

Transpose matrix A where,

$A=\begin{bmatrix} 5&4 &7 \ 8&4 &2 \ 0&9 &5 \end{bmatrix}$

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Answer

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: $\begin{bmatrix} 1& 2&3 \ 4&5 &6 \ 7&8 &9 $\end{bmatrix}^{T}$ = \begin{bmatrix} 1&4 &7 \ 2&5 &8 \ 3&6 &9 \end{bmatrix}$ ie. column 1 become row 1, column 2 becomes row 2, etc.

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Question

$\begin{bmatrix} 2&5 \ 4& 5\ 3&2 \ 0&1 $\end{bmatrix}^{T}$=$

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Answer

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: $\begin{bmatrix} 1& 2&3 \ 4&5 &6 \ 7&8 &9 $\end{bmatrix}^{T}$ = \begin{bmatrix} 1&4 &7 \ 2&5 &8 \ 3&6 &9 \end{bmatrix}$ ie. column 1 become row 1, column 2 becomes row 2, etc.

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Question

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

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Answer

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: $\begin{bmatrix} 1& 2&3 \ 4&5 &6 \ 7&8 &9 $\end{bmatrix}^{T}$ = \begin{bmatrix} 1&4 &7 \ 2&5 &8 \ 3&6 &9 \end{bmatrix}$ ie. column 1 become row 1, column 2 becomes row 2, etc.

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$\begin{bmatrix} 2&10 &54 \ -5& 6& 17\ 25& 4& 0\ 2& 12&-4 $\end{bmatrix}^{T}$=$

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Answer

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: $\begin{bmatrix} 1& 2&3 \ 4&5 &6 \ 7&8 &9 $\end{bmatrix}^{T}$ = \begin{bmatrix} 1&4 &7 \ 2&5 &8 \ 3&6 &9 \end{bmatrix}$ ie. column 1 become row 1, column 2 becomes row 2, etc.

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Question

$\begin{bmatrix} 7&8 \ 0&2 $\end{bmatrix}^{T}$=$

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Answer

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: $\begin{bmatrix} 1& 2&3 \ 4&5 &6 \ 7&8 &9 $\end{bmatrix}^{T}$ = \begin{bmatrix} 1&4 &7 \ 2&5 &8 \ 3&6 &9 \end{bmatrix}$ ie. column 1 become row 1, column 2 becomes row 2, etc.

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Question

$\begin{bmatrix} 5\ -8\ 10 $\end{bmatrix}^{T}$=$

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Answer

Transposing a matrix simply means to make the columns of the original matrix the rows in the transposed matrix. Example: $\begin{bmatrix} 1& 2&3 \ 4&5 &6 \ 7&8 &9 $\end{bmatrix}^{T}$ = \begin{bmatrix} 1&4 &7 \ 2&5 &8 \ 3&6 &9 \end{bmatrix}$ ie. column 1 become row 1, column 2 becomes row 2, etc.

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Question

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

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Answer

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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Question

Find the transpose of matrix A.

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

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Answer

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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Question

Find the transpose of matrix A.

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

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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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Question

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

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Answer

$\begin{align}&$\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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Question

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

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Answer

$\begin{align}&$\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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Question

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

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Answer

$\begin{align}&$\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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Question

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

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Answer

$\begin{align}&$\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}-13&-12&15&1&-18\0&3&-17&-1&-13\end{bmatrix}\&A^{T}$=\begin{bmatrix}-13&0\-12&3\15&-17\1&-1\-18&-13\end{bmatrix}\end{align}$

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Question

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

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Answer

$\begin{align}&$\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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Question

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

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Answer

$\begin{align}&$\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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Question

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

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Answer

$\begin{align}&$\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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Question

$\begin{align}&$\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}$\end{align}$

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Answer

$\begin{align}&$\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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Question

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

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Answer

$\begin{align}&$\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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Question

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

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Answer

$\begin{align}&$\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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