# 11.6 Solving systems with gaussian elimination  (Page 2/13)

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## Writing a system of equations from an augmented matrix form

Find the system of equations from the augmented matrix.

When the columns represent the variables $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ $y,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z,$

Write the system of equations from the augmented matrix.

$\left[\begin{array}{ccc}1& -1& \text{\hspace{0.17em}}\text{\hspace{0.17em}}1\\ 2& -1& \text{\hspace{0.17em}}\text{\hspace{0.17em}}3\\ 0& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1& \text{\hspace{0.17em}}\text{\hspace{0.17em}}1\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}|\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}5\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}1\\ -9\end{array}\right]$

$\begin{array}{c}x\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z=5\\ 2x\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}3z=1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z=-9\end{array}$

## Performing row operations on a matrix

Now that we can write systems of equations in augmented matrix form, we will examine the various row operations    that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows.

Performing row operations on a matrix is the method we use for solving a system of equations. In order to solve the system of equations, we want to convert the matrix to row-echelon form    , in which there are ones down the main diagonal    from the upper left corner to the lower right corner, and zeros in every position below the main diagonal as shown.

We use row operations corresponding to equation operations to obtain a new matrix that is row-equivalent    in a simpler form. Here are the guidelines to obtaining row-echelon form.

1. In any nonzero row, the first nonzero number is a 1. It is called a leading 1.
2. Any all-zero rows are placed at the bottom on the matrix.
3. Any leading 1 is below and to the right of a previous leading 1.
4. Any column containing a leading 1 has zeros in all other positions in the column.

To solve a system of equations we can perform the following row operations to convert the coefficient matrix    to row-echelon form and do back-substitution to find the solution.

1. Interchange rows. (Notation: $\text{\hspace{0.17em}}{R}_{i}\text{\hspace{0.17em}}↔\text{\hspace{0.17em}}\text{\hspace{0.17em}}{R}_{j}$ )
2. Multiply a row by a constant. (Notation: $\text{\hspace{0.17em}}c{R}_{i}$ )
3. Add the product of a row multiplied by a constant to another row. (Notation: $\text{\hspace{0.17em}}{R}_{i}+c{R}_{j}\right)$

Each of the row operations corresponds to the operations we have already learned to solve systems of equations in three variables. With these operations, there are some key moves that will quickly achieve the goal of writing a matrix in row-echelon form. To obtain a matrix in row-echelon form for finding solutions, we use Gaussian elimination, a method that uses row operations to obtain a 1 as the first entry so that row 1 can be used to convert the remaining rows.

## Gaussian elimination

The Gaussian elimination    method refers to a strategy used to obtain the row-echelon form of a matrix. The goal is to write matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ with the number 1 as the entry down the main diagonal and have all zeros below.

The first step of the Gaussian strategy includes obtaining a 1 as the first entry, so that row 1 may be used to alter the rows below.

Given an augmented matrix, perform row operations to achieve row-echelon form.

1. The first equation should have a leading coefficient of 1. Interchange rows or multiply by a constant, if necessary.
2. Use row operations to obtain zeros down the first column below the first entry of 1.
3. Use row operations to obtain a 1 in row 2, column 2.
4. Use row operations to obtain zeros down column 2, below the entry of 1.
5. Use row operations to obtain a 1 in row 3, column 3.
6. Continue this process for all rows until there is a 1 in every entry down the main diagonal and there are only zeros below.
7. If any rows contain all zeros, place them at the bottom.

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