Elementary Row Operations
1. (Replacement) Replace one row by the sum of itself and a multiple of another row.2. (Interchange) Interchange two rows3. (Scaling) Multiply all entries in a row by a nonzero constant
Elementary row operations are transformations that can be performed on a matrix to make it easier to perform certain calculations, or to put the matrix into a specific form. There are three types of elementary row operations, as follows:
1. Swapping Rows: This operation allows for two rows of a matrix to be switched. This operation is written as: Ri ↔ Rj (where i and j are the row numbers to be swapped).
Example:
Suppose we have a matrix A = [ 1 2 ; 3 4 ]. Interchanging the first row with the second row would result in the matrix B= [ 3 4 ; 1 2 ]
2. Scaling a Row: This operation allows for one row of a matrix to be multiplied by a nonzero constant. This operation is written as: kRi -> Ri (where k is the nonzero constant, and i is the row number).
Example:
Suppose we have a matrix A = [ 1 2 ; 3 4 ]. Scaling the first row by a factor of 5 would result in the matrix B= [ 5 10 ; 3 4 ].
3. Adding a Multiple of One Row to Another: This operation allows for one row of a matrix to be multiplied by a constant and added to another row. This operation is written as: kRi + Rj -> Rj (where i and j are the row numbers, and k is the constant multiplier).
Example:
Suppose we have a matrix A = [ 1 2 ; 3 4 ]. Adding 2 times the first row to the second row would result in the matrix B= [ 1 2 ; 5 8 ].