# Elementary Operations on a Matrix

### Elementary Row Operations:

1. Row Replacement (or Row Addition):

• Replace one row with the sum of itself and a multiple of another row.
• For row $i$ and row $j$ in a matrix $A$, ${R}_{i}={R}_{i}+k×{R}_{j}$.
2. Row Interchange:

• Interchange (swap) two rows within a matrix.
• Exchange rows $i$ and $j$ in matrix $A$.
3. Row Scaling (or Row Multiplication):

• Multiply a row by a non-zero scalar.
• For row $i$ in matrix $A$, ${R}_{i}=k×{R}_{i}$.

### Elementary Column Operations:

1. Column Replacement (or Column Addition):

• Replace one column with the sum of itself and a multiple of another column.
• For column $i$ and column $j$ in a matrix $A$, ${C}_{i}={C}_{i}+k×{C}_{j}$.
2. Column Interchange:

• Interchange (swap) two columns within a matrix.
• Exchange columns $i$ and $j$ in matrix $A$.
3. Column Scaling (or Column Multiplication):

• Multiply a column by a non-zero scalar.
• For column $i$ in matrix $A$, ${C}_{i}=k×{C}_{i}$.

### Example:

Consider a matrix $A$:

$A=\left[\begin{array}{cc}2& 3\\ 5& 4\end{array}\right]$
1. Row Replacement: ${R}_{2}={R}_{2}-5×{R}_{1}$
$\left[\begin{array}{cc}2& 3\\ 5& 4\end{array}\right]\to \left[\begin{array}{cc}2& 3\\ -23& -11\end{array}\right]$
1. Row Interchange: Swap rows 1 and 2
$\left[\begin{array}{cc}2& 3\\ 5& 4\end{array}\right]\to \left[\begin{array}{cc}5& 4\\ 2& 3\end{array}\right]$
1. Row Scaling: ${R}_{1}=\frac{1}{2}×{R}_{1}$
$\left[\begin{array}{cc}2& 3\\ 5& 4\end{array}\right]\to \left[\begin{array}{cc}1& \frac{3}{2}\\ 5& 4\end{array}\right]$

### Example:

Consider a matrix $A$:

$A=\left[\begin{array}{cc}2& 3\\ 5& 4\end{array}\right]$
1. Column Replacement: ${C}_{2}={C}_{2}-3×{C}_{1}$
$\left[\begin{array}{cc}2& 3\\ 5& 4\end{array}\right]\to \left[\begin{array}{cc}2& -7\\ 5& -11\end{array}\right]$
1. Column Interchange: Swap columns 1 and 2
$\left[\begin{array}{cc}2& 3\\ 5& 4\end{array}\right]\to \left[\begin{array}{cc}3& 2\\ 4& 5\end{array}\right]$
1. Column Scaling: ${C}_{1}=\frac{1}{2}×{C}_{1}$
$\left[\begin{array}{cc}2& 3\\ 5& 4\end{array}\right]\to \left[\begin{array}{cc}1& 3\\ 5& 4\end{array}\right]$

### Properties and Significance:

• Elementary operations do not change the solutions of a system of linear equations represented by the matrix.

• They are fundamental in various matrix operations, including row reduction, finding inverses, and solving systems of equations.