# nxn Transpose Matrix Calculator

Matrix Type
Matrix A

Transpose Matrix

CALCULATE

$n\times n$ Transpose Matrix calculator calculates a transpose matrix of a matrix $A$ with real elements. It is an online math tool specially programmed to convert the matrix $A$ to transpose matrix $A^T$ by interchanging rows and columns of matrix $A$. This calculator is applicable for matrices $3\times 3$, $3\times 2$, $3\times 1$, $2\times 3$, $2\times 2$, $2\times 1$, $1\times 3$, $1\times 2$. Select the appropriate calculator from the list of eight.
It is necessary to follow the next steps:

1. Enter elements of the matrix in the box. Elements of matrices must be real numbers.
2. Press the "CALCULATE" button to make the computation;
3. $n\times n$ transpose matrix calculator will give the matrix which represents the transpose matrix of the given matrix.
Input : A matrix with real elements of dimension $m \times n$;
Output : A matrix with real elements of dimension $n \times m$.

Transpose Matrix Formula
$3\times 3$ Transpose Matrix Formula: The transpose matrix of the matrix $A=\left( \begin{array}{ccc} a & b & c \\ d& e & f \\ g & h & i \\ \end{array} \right)$ is determined by the following formula $$A^T=\left( \begin{array}{ccc} a & d & g \\ b& e & h \\ c & f & i \\ \end{array} \right)$$ $3\times 2$ Transpose Matrix Formula: The transpose matrix of the matrix $A=\left( \begin{array}{cc} a & b \\ c& d \\ e & f \\ \end{array} \right)$ is determined by the following formula $$A^T=\left( \begin{array}{ccc} a & c & e \\ b& d & f \\ \end{array} \right)$$ $3\times 1$ Transpose Matrix Formula: The transpose matrix of the matrix $A=\left( \begin{array}{c} a \\ b \\ c \\ \end{array} \right)$ is determined by the following formula $$A^T=\left( \begin{array}{ccc} a & b & c \\ \end{array} \right)$$ $2\times 3$ Transpose Matrix Formula: The transpose matrix of the matrix $A=\left( \begin{array}{ccc} a & b & c \\ d& e & f \\ \end{array} \right)$ is determined by the following formula $$A^T=\left( \begin{array}{cc} a & d \\ b& e \\ c & f \\ \end{array} \right)$$ $2\times 2$ Transpose Matrix Formula: The transpose matrix of the matrix $A=\left( \begin{array}{cc} a & b \\ c& d \\ \end{array} \right)$ is determined by the following formula $$A^T=\left( \begin{array}{cc} a & c \\ b& d \\ \end{array} \right)$$ $2\times 1$ Transpose Matrix Formula: The transpose matrix of the matrix $A=\left( \begin{array}{c} a \\ b \\ \end{array} \right)$ is determined by the following formula $$A^T=\left( \begin{array}{cc} a & b \\ \end{array} \right)$$ $1\times 3$ Transpose Matrix Formula: The transpose matrix of the matrix $A=\left( \begin{array}{ccc} a &b&c \\ \end{array} \right)$ is determined by the following formula $$A^T=\left( \begin{array}{c} a \\ b\\ c\\ \end{array} \right)$$ $1\times 2$ Transpose Matrix Formula: The transpose matrix of the matrix $A=\left( \begin{array}{cc} a &b \\ \end{array} \right)$ is determined by the following formula $$A^T=\left( \begin{array}{c} a \\ b\\ \end{array} \right)$$

## What is Transpose Matrix?

The transpose matrix, denoted by $A^T$, is a new matrix whose rows are the columns of the original matrix $A$ and the columns of the new matrix is the rows of the matrix $A$. The superscript "T" means "transpose". For instance, the transpose of the $3\times 3$ matrix $A=\left( \begin{array}{ccc} a & b & c \\ d& e & f \\ g & h & i \\ \end{array} \right)$ is $$A^T=\left( \begin{array}{ccc} a & b & c \\ d& e & f \\ g & h & i \\ \end{array} \right)^T=\left( \begin{array}{ccc} a & d & g \\ b& e & h \\ c & f & i \\ \end{array} \right)$$ In other words, the element $a_{ij}$ of the original matrix $A$ becomes element $a_{ji}$ in the transposed matrix $A^T$. Usually, we find the transpose of square matrices, but non-square matrices can be also transposed. For example, $$\left(\begin{array}{cccc} a & b & c&d \\ e& f & g&h \\ \end{array} \right)^T=\left(\begin{array}{cc} a & e \\ b& f \\ c & g \\ d&h\\ \end{array} \right)$$ Therefore, if $A = (a_{ij})_{m\times n}$, then $A^T = (a_{ji})_{m\times n}$. There are some properties of transpose matrices:

1. $(A^T)^T=A$;
2. $(A\pm B)^T=A^T\pm B^T$;
3. $(kA)^T=kA^T,$ \quad $k\in \mathbb{R}$;
4. $(AB)^T=B^TA^T.$

### How to Find Transpose Matrix?

The transpose matrix of a square matrix is a new matrix which flips a matrix over its main diagonal. This means it switches the rows and columns. To find the transpose of any matrix $A$ follow one of the steps:

• if matrix $A$ is a square matrix, reflect $A$ over its main diagonal;
• write the rows of $A$ as the columns of $A^T$;
• write the columns of $A$ as the rows of $A^T$.
The $n\times n$ Transpose Matrix work with steps shows the complete step-by-step calculation for finding a transpose of $3\times 3$, $3\times 2$, $3\times 1$, $2\times 3$, $2\times 2$, $2\times 1$, $1\times 3$, $1\times 2$ matrices using the matrix transpose formula. For any other matrices, just supply real numbers as elements of the matrix and click on the Generate Work button. The grade school students use this $n\times n$ Transpose Matrix Calculator to generate the work, verify the results of transpose matrix derived by hand or do their homework problems efficiently.

### Real World Problems of Inverse Matrix

Recall, that dot product between two vectors $\vec a$ and $\vec b$ is $$\vec a\cdot\vec b=|\vec a|\; |\vec b|\cos\theta$$ where $\theta$ is the angle between these vectors. This product can be written as $\vec a^T\vec b$.
A digital image can be represented by matrices. For example, if we consider the image $A$ as a matrix, then the image $B$ corresponds to the transposed matrix of $A$.

### Practice Problems to Find Transpose Matrix

Practice Problem 1:
Find the transpose matrix of the matrix $\left( \begin{array}{ccc} 1 & 7 &5\\ -1 &3 &6\\ \end{array} \right)$.

Practice Problem 2:
Let $\vec a$ and $\vec b$ be two three-dimensional vectors $\vec a=(1,3,4)$ and $\vec b=(-3,-6,3)$. Find ${\vec a}^T{\vec b}$.

The $n\times n$ inverse matrix calculator, formula, practice and real world problems would be very useful for grade school students (K-12 education) to understand the concept of transpose matrix and inverse matrix. This concept will be helpful in solving linear algebra problems.