Question

What is the elementary transformation of matrices ?

Answer 1

Elementary transformation of matrices.

In order to transform a matrix into a specific required form, some operations are done, known as elementary operations or transformation of matrices. There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or transformations.

The operations (transformations) on a matrix:

• Interchanging two rows, multiplication a row by either a nonzero value. Denoted by $R_i\leftrightarrow R_j$or $C_i\leftrightarrow C_j$ i.e. interchange of $i^{th}\mathrm{and}{j}^{th}$ row or column respectively.

$A=\left[\begin{array}{ccc}1& -2& 2\\ -1& 1& 3\\ 1& -1& -4\end{array}\right]$

Perform the operation which interchange rows $R_2\leftrightarrow R_1$on $A$:

$A\text{'}=\left[\begin{array}{ccc}-1& 1& 3\\ 1& -2& 2\\ 1& -1& -4\end{array}\right]$

• The addition to the elements of any row or column, the corresponding elements of any other row or column are multiplied by any non-zero number. denoted by$R_i\to R_i+k{R}_j$ i.e.the multiplication of each element of the$i^{th}$ row by k, where k ≠ 0. i.e. the addition to the elements of $i^{th}$ row, the corresponding elements of$j^{th}$ row multiplied by $k$

Let us consider a matrix $A$:

$A=\left[\begin{array}{ccc}1& -2& 2\\ -1& 1& 3\\ 1& -1& -4\end{array}\right]$

Perform the operation $R_2\to R_2+R_1$on $A$:

$A\text{'}=\left[\begin{array}{ccc}1& -2& 2\\ 0& -1& 5\\ 1& -1& -4\end{array}\right]$

• The multiplication of the elements of any row or column by a non-zero number. Denoted by $R_i\to k{R}_i$ or $C_i\to k{C}_i$ i.e..the multiplication of each element of the $i^{th}$ row by $k$, where $k\ne 0$

Let us consider a matrix $A$:

$A=\left[\begin{array}{ccc}1& -2& 2\\ -1& 1& 3\\ 1& -1& -4\end{array}\right]$

Perform the operation $C_2\to 3C_2$ on $A$

$A\text{'}=\left[\begin{array}{ccc}1& -6& 2\\ -1& 3& 3\\ 1& -3& -4\end{array}\right]$