Description of matrices and elementary row operations
There are three types of elementary matrix row operations, corresponding to the operations that apply to equations to eliminate variable
Adding a multiple of one row to another row
Multiplying of a row by a non-zero scalar
Interchange of two rows
These operations can be done manually, but also by matrices multiplication with a given matrix and some modified identity matrix. See the three examples below.
Adding a multiple of one row to another
Placing \(k\) in the second column of row 3 of the identity matrix
then multiplying the matrices.
This has k-times the values of corresponding elements of row 2 added to those of row 3 of the matrix.
\(\displaystyle \left[\matrix{1&0&0\\0&1&0\\0&k&1}\right]˙ \left[\matrix{a&b&c\\d&e&f\\g&h&i}\right]= \left[\matrix{a&b&c\\d&e&f\\kd+g&ke+h&kf+i}\right]\)
The value of the determinant in the result is identical to the value of the source matrix \(A\)
Multiplying a row by a non-zero scalar:
\(\displaystyle \left[\matrix{1&0&0\\0&k&0\\0&0&1}\right] · \left[\matrix{a&b&cd\\d&e&f\\g&h&i}\right]= \left[\matrix{a&b&c\\kd&ke&kf\\g&h&i}\right]\)
The value of the determinant in the result is \(k\) times the value of the source matrix \(A\)
Interchanging two rows
\(\displaystyle \left[\matrix{1&0&0\\0&0&1\\0&1&0}\right]˙ \left[\matrix{a&b&c\\d&e&f\\g&h&i}\right]= \left[\matrix{a&b&c\\g&h&i\\s&e&f}\right]\)
The value of the determinant in the result is identical to the value of the source matrix \(A\)