What makes an augmented matrix inconsistent

There are three valid operations we can perform on our system of equations:. We can make our life easier by extracting only the numbers, and putting them in a box:. This is called an augmented matrix. In this notation, our three valid ways of manipulating our equations become row operations :. Of course this does not mean that the second row is equal to the second row minus twice the first row.

Instead it means that we are replacing the second row with the second row minus twice the first row. This kind of syntax is used frequently in computer programming when we want to change the value of a variable. The process of doing row operations to a matrix does not change the solution set of the corresponding linear equations! Indeed, the whole point of doing these operations is to solve the equations using the elimination method. Two matrices are called row equivalent if one can be obtained from the other by doing some number of row operations.

So the linear equations of row-equivalent matrices have the same solution set. In the previous subsection we saw how to translate a system of linear equations into an augmented matrix. A pivot is the first nonzero entry of a row of a matrix in row echelon form. A matrix in row-echelon form is generally easy to solve using back-substitution. For example,. See this example. Determine all possibilities for the solution set of the system of linear equations described below.

First note that the system is homogeneous and hence it is consistent. Thus if the system has a nontrivial […] Solving a System of Linear Equations Using Gaussian Elimination Solve the following system of linear equations using Gaussian elimination. Remark: a null space is also called a kernel. The solutions will be given after completing all problems.

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Learn more. Explain why this augmented matrix is consistent for all h? Ask Question. Asked 6 years, 9 months ago. Active 6 years, 9 months ago. Viewed 2k times. Add a comment. Active Oldest Votes. Daniel W. Farlow Daniel W.

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