ELEMENTARY ROW OPERATIONS OF A MATRIX
Here we will discuss about the elementry row operations. The whole data was carefully typed. If you found any mistake then comment please.
Earn Money online without investment Earn Money online from home without investment
Consider the identity matrix
I_3=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]The matrices
E_1=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}\right], E_2=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 1
\end{array}\right],E_3=\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & -5 & 1
\end{array}\right]have been obtained from [katex]I_3[/katex] by the following operations.
-
- [katex]E_1[/katex] : Interchange of two rows of [katex]I_3[/katex]. (Second and third rows interchanged)
-
- [katex]E_2[/katex] : Multiply a row of [katex]I_3[/katex] by a nonzero number. (Here row two is multiplied by 3 )
-
- [katex]E_3[/katex]: Add a multiple of a row of [katex]I_3[/katex] to another row. ( -5 times row two is added to row three):
These operations on [katex]l_3[/katex] are called elementary row operations and corresponding matrices [katex]E_1, E_2, E_3[/katex] are called elementary matrices of order 3 . Now let
A=\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{12} & a_{33}
\end{array}\right]We examine the effect of premultiplying A by [katex]E_1, E_2[/katex] and [katex]E_3[/katex] respectively.
E_1 A=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}\right]\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{13}
\end{array}\right]=\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{31} & a_{12} & a_{33} \\
a_{21} & a_{22} & a_{23}
\end{array}\right]= A matrix obtained from A by interchange of rows two and three.
E_2 A=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 1
\end{array}\right]\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right]=\left[\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
3 a_{21} & 3 a_{22} & 3 a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right]= A matrix obtained from A by multiplying row two by 3 .
E_3 A=\left[\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & -5 & 1
\end{array}\right]\left[\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right]=\left[\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31}-5 a_{21} & a_{12}-5 a_{22} & a_{33}-5 a_{23}
\end{array}\right]=A matrix obtained from A by sublracting 5 times the sccond row from the third row. If we postimultiply the matrix A by
E_1^{\prime}=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}\right], E_2^{\prime}=\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 1
\end{array}\right], E_3^{\prime}=\left[\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & -5 \\
0 & 0 & 1
\end{array}\right]Then [katex]A E_1^{\prime}, A E_2^{\prime}[/katex] and [katex]A E_3^{\prime}[/katex] are the matrices obtained from A by doing similar operations on columns of A instead of rows of A. This fact can be generalized to any [katex]m \times n[/katex] matcix. That is the same operations can be performed on any arbitrary matrix
A=\left[\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1 n} \\
a_{21} & a_{22} & \cdots & a_{2 n} \\
a_{31} & a_{32} & \cdots & a_{1 n} \\
\vdots & \vdots & \cdots & \vdots \\
a_{m 1} & a_{m 2} & \cdots & a_{m n}
\end{array}\right]by premultiplying A by the [katex]m \times m[/katex] clementary matrices [katex]E_1, E_2[/katex] and [katex]E_3[/katex] obtained from [katex]I_m[/katex] by the row operations mentioned above respectively.
Definition
Definition. The following operations on a matrix A are called elementary row operations: (i) Interchange of any two rows [katex]\left(R_{i j}\right.[/katex] denotes the interchange of the th row with the jth row) (ii) Multiplication of a row of A by any nonzero real or complex number [katex]\left(k R_i\right.[/katex] denote that ith row is multiplied by [katex]k \neq 0[/katex] ). (iii) Addition of a scalar multiple of one row to another row. [katex]\left(R_j+k R_i\right)[/katex] denotes that k times row i is added to the jth row.
Note that each of these row operations on an [katex]m \times n[/katex] matrix can be effected by premultiplying [katex]A[/katex] by an [/katex]m \times m[/katex] elementary matrix.
Let A be an [katex]m \times n[/katex] matrix. An [katex]m \times n[/katex] matrix B is called row equivalent to A if B is obtained from A by performing a finite sequence of elementary row operations on A. We write [katex]B \stackrel{R}{\sim}[/katex] A to denote B is row equivalent to A.