Row Operations

Row Operations

Since matrices are really systems of equations, anything that is valid for systems is valid for matrices. Within a system of equations we can

1. Switch two equations.

2. Multiply any equation by a non-zero number.

2. Add a multiple of one equation to another.

What we will do next is solve linear systems using a different method from inverses. Instead we will use an augmented matrix. In this process, we can

1. Switch any two rows of a matrix.

2. Multiply a row by a number.

3. Add a multiple of one row to another row.

Keep in mind that we are producing different matrices when we do this, but the resulting system will be the same.

At the moment, let's look at the row operations themselves.

Define

If I want to switch the first row with the second row, I would get

If I then multiplied the second row by -1 I would get

I could then add -2 times the second row to the first and get

Row Operations using Elementary Matrices

These row operations can be accomplished just by moving the elements around or we can actually do it by using elementary matrices.

If we perform a row operation on the identity matrix, then we call that an elementary matrix.

For example, let's do the above operations on the 2X2 identity matrix

Switch the first and second rows:

Instead, take the original identity matrix and multiply the second row by -1:

Instead, take the original identity matrix and add -2 times the second row to the first and get

Each of these is an elementary matrix.

The reason why they're interesting is that we can accomplish the row operations on M by multiplying it by the appropriate elementary matrix which has had that operation done to it.

For example, the elementary matrix has had its first and second rows switched.

If we multiply we get

We can then multiply the second row of this by -1 by multiply it by

to get

then add -2 times the second row to the first by multiplying this by

to get

The point of all this is that the elementary matrices accomplish the row operations.