Linear Algebra/Row and Column Operations

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Elementary Row Operations[edit]

There are three elementary row operations:

  1. Replacement
  2. Interchanging
  3. Scaling

The following are an explanation and examples of the three different operations. In the chapter Systems of Linear Equations we have used all of these operations, except Scaling.

An important thing to remember is that all operations can be used on all matrices, not just on matrices derived from linear systems.

Replacement[edit]

Replace one row by the sum of itself and a multiple of another row. A more common paraphrase of row replacement is "Add to one row a multiple of another row."

An example is, we are given the linear system:

x_1+4x_2=3
2x_1+2x_2=4

This can be written in matrix notation, as an augmented matrix, like this

\begin{bmatrix}1 && 4 && 3 \\ 2 && 2 && 4\end{bmatrix}

Now we have decided to eliminate the x_1 term in equation 2; this can be done by adding -2 times equation 1 to equation 2

\begin{matrix}-2*[equation\ 1]: & -2x_1-8x_2=-6 \\ \underline{+[equation\ 2]:} & \underline{2x_1+2x_2=4} \\ \left[new\ equation\ 2\right]: & -6x_2=-2\end{matrix}

which gives us the matrix

\begin{bmatrix}1 && 4 && 3 \\ 0 && -6 && -2\end{bmatrix}

Interchanging[edit]

Interchange two rows.

An example is as follows. We are given the matrix

\begin{bmatrix}1 && 2 && 3 \\ 2 && 3 && 1\end{bmatrix}

Here we have performed an interchange operation on the two rows.

\begin{bmatrix}2 && 3 && 1 \\ 1 && 2 && 3\end{bmatrix}

This is a useful operation when you are trying to solve a linear system and can see that it will be easier to solve it by interchanging two rows. It is a widely used operation, even though it seems like an odd and not very useable operation.

Scaling[edit]

Multiply all entries in a row by a nonzero constant.

As an example, we are given the matrix

\begin{bmatrix}1 && 2 && 3 \\ 2 && 3 && 1\end{bmatrix}

Now a scaling operation has been performed on the first row, by multiplying by -2

\begin{bmatrix}-2 && -4 && -6 \\ 2 && 3 && 1\end{bmatrix}