Transpose of a Matrix: Overview, Questions, Preparation

In simple words, to find the transpose of matrix, just interchange the rows and columns of the matrix. It is denoted by A’ or $A^T$ . If explained in mathematical terms;

If there is a matrix A = ${\left[a_{ij}\right]}_{m\times n}$ , then the transpose of matrix A is defined as, $A^T$ = ${\left[a_{ji}\right]}_{n\times m}$  

For example, If matrix A = ${\left[\begin{array}{cc}\hfill 3\hfill & \hfill -5\hfill \\ \hfill 4\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right]}_{2\times 3}$ , then Transpose of matrix ( $A^T$ ) = ${\left[\begin{array}{ccc}\hfill 3\hfill & \hfill -1\hfill & \hfill -5\hfill \\ \hfill 1\hfill & \hfill 4\hfill & \hfill 2\hfill \end{array}\right]}_{3\times 2}$

In conclusion, the transpose of a matrix is a matrix formed by swapping the rows and columns.

• Order of Transpose Matrix: The order of the transpose is always exactly opposite to the order of the original matrix. If the matrix A is of order m x n, then the transpose of matrix ( $A^T$ ) will be of order n x m. E.g., in the above given example, the original matrix has a 2x3 order, and the transpose matrix has a 3x2 order.

Finding the transpose of a given matrix is simple. The first step is to identify the number of rows. Now, exchange all the rows with the corresponding column, thus formed matrix is transpose. How to exchange:

• 1^st Row = 1^st Column
• 2^nd Row = 2^nd Column
• 3^rd Row = 3^rd Column

Similarly; nth Row = nth Column

For Example: if A= $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$ , then

• 1^st Row $\left[\begin{array}{ccc}1& 2& 3\end{array}\right]$ becomes 1^st Column $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$
• 2^nd Row $\left[\begin{array}{ccc}4& 5& 6\end{array}\right]$ becomes 2^nd column $\left[\begin{array}{c}4\\ 5\\ 6\end{array}\right]$

The transpose A’ = $\left[\begin{array}{cc}1& 4\\ 2& 5\\ 3& 6\end{array}\right]$

You can read the definition of transpose as described in NCERT Textbook for class 11 mathematics.

“Definition 3: If A = $\left[a_{ij}\right]$  be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. Transpose of the matrix A is denoted by A′ or ${(A}^T)$ . In other words, if   $A={\left[a_{ij}\right]}_{m\times n}$ , then $A\text{'}={\left[a_{ji}\right]}_{n\times m}$ .”

You can transpose of check various types of matrices below.

• Transpose of Row and Column Matrices

Transpose of row matrix becomes a column matrix:

If a Row matrix A = $\left[\begin{array}{ccc}a& b& c\end{array}\right]$ , the transpose of matrix A’ = $\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$ , which is a column matrix.

Transpose of a Column matrix becomes a Row matrix:

If a row matrix A = $\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$ , the transpose of matrix A’ = $\left[\begin{array}{ccc}a& b& c\end{array}\right]$ , which is a row matrix.

• Transpose of a Square Matrix

The transpose of a square matrix remains the same order matrix, just the rows are exchanged with columns. For example:

If a matrix A= $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ then the transpose formed by exchanging the rows and columns; $A^{\text{'}}=\left[\begin{array}{ccc}1& 4& 7\\ 2& 5& 8\\ 3& 6& 9\end{array}\right]$

Note: An Important point in the transpose of square matrices is that diagonal elements remain unchanged. As in the above example, diagonal elements [1 5 9], remain the same.

• Transpose of a Diagonal Matrix

As we know that diagonal matrices are a special type of matrix having all the non-diagonal elements as zero. The transpose of the diagonal matrix remains as it is, which means for diagonal matrix A= A’.

For Example:
If there is a diagonal matrix $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 5& 0\\ 0& 0& 9\end{array}\right]$ , then the transpose will be $A\text{'}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 5& 0\\ 0& 0& 9\end{array}\right]$ , You can see both are the same. So, A = A’ for a diagonal matrix.

• Transpose of Symmetric and Skew-symmetric Matrix

The symmetric matrices are specific matrices for which the transpose of a symmetric matrix remains the same as the original matrix. For symmetric matrices; A = A’.  For example

A symmetric matrix $A=\left[\begin{array}{ccc}1& -1& 3\\ -1& 5& 0\\ 3& 0& 9\end{array}\right]$ , Find the transpose of this matrix by exchanging the rows with columns: $A\text{'}=\left[\begin{array}{ccc}1& -1& 3\\ -1& 5& 0\\ 3& 0& 9\end{array}\right]$ . Both symmetric matrix and its transpose are exactly same.

Skew symmetric matrices are defined such that their transpose will be equal to the negative of the given matrix. That means A = $A=-A^{\text{'}}or-A=A\text{'}$ for skew symmetric matrix.

For example, Let’s assume a skew-symmetric matrix $A=\left[\begin{array}{ccc}0& -2& 3\\ 2& 0& 1\\ -3& -1& 0\end{array}\right]$

The transpose of matrix A: $A\text{'}=\left[\begin{array}{ccc}0& 2& -3\\ -2& 0& -1\\ 3& 1& 0\end{array}\right]$

The negative of transpose will be $-A\text{'}=\left[\begin{array}{ccc}0& -2& 3\\ 2& 0& 1\\ -3& -1& 0\end{array}\right]$

• The transpose of the transpose of a matrix will be the same as the matrix A: (A')' = A.
• The transpose is distributive over addition. The transpose of the sum of two matrices will be equal to the sum of the transposes of both matrices: (A + B)' = A' + B'.
• The transpose of a matrix multiplied by any scalar will be the same as the transpose multiplied by the scalar: (kA)' = kA'.
• The transpose of the product of two matrices will be equal to the product of the individual transposes of both matrices in reverse order: (AB)' = B'A'.
• For a symmetric matrix, the transpose and original matrix are equal: A = A'.
• For a skew-symmetric matrix, the negative of the transpose and the original matrix are equal: A = -A'.
• Determinant of transpose of matrix (A') will be equal to the determinant of the matrix: |A|=|A’|.

• Transpose is widely used to find the adjoint and inverse of the matrices.
• Transpose also helps in solving the system of linear equations through transposing the co-efficient matrix.
• Transpose of a matrix is used in coordinate geometry to transform the coordinate points, for rotations, and reflections.
• Transpose of matrices have various real life applications such as cryptography, computer graphics, 3D model generation, Data analysis, and more.