Transpose of a Matrix

The transpose of a matrix is one of the most common methods used for matrix transformation in matrix concepts across linear algebra.

Definition

The transpose of a matrix is obtained by interchanging its rows into columns or columns into rows. The transpose of the matrix is denoted by using the letter $T$ in the superscript of the given matrix. For example, if $A$ is the given matrix, then the transpose of the matrix is represented by $A^{'}$ or$A^{T}$.

Transpose of a matrix

$A={\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix}}_{2\times 3}$ $A^{T}={\begin{bmatrix}a_{11}&a_{21}\\a_{12}&a_{22}\\a_{13}&a_{23}\end{bmatrix}}_{3\times 2}$

Example

Find the transpose of $A={\begin{bmatrix}3&4&5\\6&7&8\end{bmatrix}}_{2\times 3}$

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

Properties of transpose of the matrices

Let us take two matrices $A$ and $B$ which have equal order. Some properties of the transpose of a matrix are given below:

Transpose of the Transpose Matrix

If we take the transpose of the transpose matrix, the matrix obtained is equal to the original matrix.

Hence, for a matrix $A$, $(A^{'})^{'}=A$

Example

If $A={\begin{bmatrix}3&4&5\\6&7&8\end{bmatrix}}$ then $A^{'}={\begin{bmatrix}3&6\\4&7\\5&8\end{bmatrix}}$

$(A^{'})^{'}={\begin{bmatrix}3&4&5\\6&7&8\end{bmatrix}}=A$

Addition Property of Transpose

Transpose of an addition of two matrices $A$ and $B$ obtained will be equal to the sum of the transpose of individual matrices $A$ and $B$

Hence $(A+B)^{'}=A^{'}+B^{'}$

Example

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

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

$A^{'}+B^{'}={\begin{bmatrix}3+1&6+4\\4+2&7+5\\5+3&8+6\end{bmatrix}}={\begin{bmatrix}4&10\\6&12\\8&14\end{bmatrix}}$

$A+B={\begin{bmatrix}3+1&4+2&5+3\\6+4&7+5&8+6\end{bmatrix}}={\begin{bmatrix}4&6&8\\10&12&14\end{bmatrix}}$

$(A+B)^{'}={\begin{bmatrix}4&10\\6&12\\8&14\end{bmatrix}}=A^{'}+B^{'}$

Multiplication by Constant

If a matrix is multiplied by a constant and its transpose is taken, then the matrix obtained is equal to the transpose of the original matrix multiplied by that constant.

Hence $(kA)^{'}=kA^{'}$ where $k$ is a constant

Example

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

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

$kA^{'}=k{\begin{bmatrix}2&1\\-1&2\\2&4\end{bmatrix}}={\begin{bmatrix}2k&1k\\-1k&2k\\2k&4k\end{bmatrix}}$

$kA=k{\begin{bmatrix}2&-1&2\\1&2&4\end{bmatrix}}={\begin{bmatrix}2k&-1k&2k\\1k&2k&4k\end{bmatrix}}$

$(kA)^{'}={\begin{bmatrix}2k&1k\\-1k&2k\\2k&4k\end{bmatrix}}=kA^{'}$

Multiplication Property of Transpose

Transpose of the product of two matrices is equal to the product of transpose of the two matrices in reverse order.

Hence $(AB)^{'}=B^{'}A^{'}$,

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

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

$B^{'}A^{'}={\begin{bmatrix}1\times 3+4\times 4&1\times 6+4\times 7\\2\times 3+5\times 4&2\times 6+5\times 7\\\end{bmatrix}}={\begin{bmatrix}3+16&6+28\\6+20&12+35\\\end{bmatrix}}={\begin{bmatrix}19&34\\26&47\\\end{bmatrix}}$

$AB={\begin{bmatrix}3\times 1+4\times 4&3\times 2+4\times 5\\6\times 1+7\times 4&6\times 2+7\times 5\\\end{bmatrix}}={\begin{bmatrix}3+16&6+20\\6+28&12+35\\\end{bmatrix}}={\begin{bmatrix}19&26\\34&47\\\end{bmatrix}}$

$(AB)^{'}={\begin{bmatrix}19&34\\26&47\end{bmatrix}}=B^{'}A^{'}$