91Ó°ÊÓ

In mathematics, a system of linear equations is a collection of one or more linear equations involving the same set of variables. For example, the following is a system of linear equations in the variables x and y:
\[ \begin{aligned} &x + y = 4 \ &x - y = 2 \end{aligned} \]
The solution to a system of linear equations is the set of values for the variables that satisfies all the equations simultaneously. In the above example, we are looking for numbers x and y that make both equations true when substituted into them. Systems like this can have one solution, no solutions, or infinitely many solutions. Gauss-Jordan row reduction is one method used to find a unique solution if one exists. It simplifies the system to a point where the solution, if it exists, can be read directly from it.

An augmented matrix is a matrix that contains the coefficients of a system of linear equations as well as the constants from the right-hand side of the equations. It's like a shorthand that keeps all the important numbers together without the variables or the equals signs.
For the example system given above, the augmented matrix would be constructed as follows:
\[\begin{bmatrix}1 & 1 & | & 4 \1 & -1 & | & 2\end{bmatrix}\]
The vertical bar separates the coefficients of the variables from the constants, making it visually clearer where the equation 'splits'. Using the augmented matrix, we can apply row operations to find the solution to the system without rewriting the equations after each operation.

Row operations are the tools we use to perform Gauss-Jordan row reduction on a matrix. There are three types of row operations we can use:

• Swapping two rows,
• Multiplying a row by a non-zero scalar,
• Adding or subtracting a multiple of one row to another row.

In the exercise at hand, we used the third type of operation to make the matrix closer to the identity matrix form. The steps included subtracting the first row from the second to get zeros below the leading one in the first row, and then multiplying the second row by \(-\frac{1}{2}\) to get a leading one in the second row.
After that, to get a zero above this new leading one, we subtracted the second row from the first. These operations simplify the matrix while preserving the relationship between the variables and their solutions.

In the context of solving systems of linear equations, the identity matrix serves as the goal in the Gauss-Jordan reduction process. An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. The size of the identity matrix corresponds to the number of variables in the system.
For a two-variable system like ours, an identity matrix looks like this:
\[\begin{bmatrix}1 & 0 \0 & 1\end{bmatrix}\]
After completing Gauss-Jordan row reduction, if you can transform the left side of the augmented matrix into an identity matrix, the right side will display the solution to the original system. The process transforms complicated systems into simpler, equivalent systems that clearly show the solution.