# Transpose of a Matrix

The transpose of a matrix involves switching its rows with its columns. If $A$ is an $m×n$ matrix, its transpose ${A}^{T}$ is an $n×m$ matrix.

### Representation:

Given matrix $A$:

$A=\left[\begin{array}{cccc}{a}_{11}& {a}_{12}& \cdots & {a}_{1n}\\ {a}_{21}& {a}_{22}& \cdots & {a}_{2n}\\ & & \ddots & \\ {a}_{m1}& {a}_{m2}& \cdots & {a}_{mn}\end{array}\right]$

The transpose ${A}^{T}$ of matrix $A$ is:

${A}^{T}=\left[\begin{array}{cccc}{a}_{11}& {a}_{21}& \cdots & {a}_{m1}\\ {a}_{12}& {a}_{22}& \cdots & {a}_{m2}\\ & & \ddots & \\ {a}_{1n}& {a}_{2n}& \cdots & {a}_{mn}\end{array}\right]$

$\left({A}^{T}{\right)}^{T}=A$: The transpose of a transpose matrix is the original matrix.

### Properties:

1. Symmetric Matrices: A matrix is symmetric if and only if it is equal to its transpose: $A={A}^{T}$. Symmetric matrices have properties useful in various mathematical applications.

• $\left(cA{\right)}^{T}=c{A}^{T}$ where $c$ is a scalar.
• $\left(A+B{\right)}^{T}={A}^{T}+{B}^{T}$ where $A$ and $B$ are matrices of the same size.

### Practical Applications:

1. Solving Systems of Equations: Transposing matrices can aid in solving systems of linear equations by using methods like Gaussian elimination.

2. Matrix Operations: In matrix multiplication, the transpose plays a crucial role:

• $\left(AB{\right)}^{T}={B}^{T}\cdot {A}^{T}$ (Order reversal in multiplication)
3. Data Manipulation: In data science and machine learning, transposing matrices is commonly used to reorganize data for better analysis or to fit it into specific algorithms.

### Example:

Given matrix $A$:

$A=\left[\begin{array}{cc}3& 1\\ 5& 2\end{array}\right]$

The transpose of matrix $A$, denoted ${A}^{T}$, is:

${A}^{T}=\left[\begin{array}{cc}3& 5\\ 1& 2\end{array}\right]$