Matrix Row Operations
There are 3 basic operations used on the rows of amatrixwhen you are using the matrix tosolve a system of linear equations.目标通常是让左的一部分matrix to look like theidentity matrix.
The three operations are:
- Switching Rows
- Multiplying a Row by a Number
- Adding Rows
Switching Rows
You can switch the rows of a matrix to get a new matrix.
In the example shown above, we move Rowto Row, Rowto Row, and Rowto Row.(The reason for doing this is to get ain the top left corner.)
Multiplying a Row by a Number
You can multiply any row by a number. (This means multiplying every entry in the row by the same number.)
In this example, we have multiplied Rowof the matrix by.(This gives us thewe need in Row, Column.)
Adding Rows
You can also add two rows together, and replace a row with the result.
For example, in the matrix that resulted in the last example, we can add rowsandtogether, entry by entry:
Then, we replace Rowwith the result.
Adding Multiples of Rows
We said there were only three operations, and there are. But by using the last two operations in combination, we can add whole multiples of rows to other rows, to make things go faster.
Back up a step, so we have the matrix:
Now instead of just adding Row+ Row, add Row:
Then replace Rowwith the result.
This way, we get ain Row, Column.
We can do this again to get ain Row, Column.Here, we multiply Rowby, add it Row, and replace Rowwith the result.
We'll show a few more steps, to get theidentity matrix on the left (and thus solve the system).
The next step is to addto get ain Row, Column.
Next, we need a zero in Row, Column.
The last step is just an application of the second operation, multiplying a row by a number.
We now have the solution as ordered triple.
Important Note:If the equations represented by your original matrix represent identical or parallel lines, you will not be able to get the identity matrix using these row operations. In this case, the solution either does not exist or there are infinitely many solutions to the system.