CALCULATE

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:

- Enter elements of the matrix in the box. Elements of matrices must be real numbers.
- Press the "
**CALCULATE**" button to make the computation; - $n\times n$ transpose matrix calculator will give the matrix which represents the transpose matrix of the given 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:

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

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 $A^T$.

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 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.