When solving a system of equations using a matrix, the system can be identified as an inconsistent system if row operations result in a row where all entries to the left of the vertical line are $$0$$, but the entry on the right is a non-zero number. When this specific matrix row is translated back into an equation, it produces a mathematically false statement (such as $$0 = 1$$). Just as when solving a system using other algebraic methods, this false statement tells us that the system is an inconsistent system and there is absolutely no solution.

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To solve a system of linear equations using matrices, one must systematically transform the initial augmented matrix into row-echelon form. Begin by writing the augmented matrix representing the system. Then, apply row operations to force the entry in row $$1$$, column $$1$$ to be $$1$$, followed by getting zeros in the remainder of column $$1$$ below that $$1$$. Continue this structured process—for instance, forcing the entry in row $$2$$, column $$2$$ to be $$1$$—until the entire matrix reaches row-echelon form. Finally, translate the matrix back into a system of equations, use back-substitution to determine any remaining variables, express the final solution as an ordered pair or triple, and verify the result against the original equations.

Solving a System of Equations Using Matrices

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OpenStax Intermediate Algebra

Apply the matrix method to solve a system of three linear equations, such as: $$ \left\{\begin{array}{l} 3x + 8y + 2z = -5 \\ 2x + 5y - 3z = 0 \\ x + 2y - 2z = -1 \end{array}\right. $$ First, write the augmented matrix for the system. Next, systematically apply row operations to achieve row-echelon form, ensuring the diagonal entries are $$1$$ and the entries below them in their respective columns are $$0$$. Once the matrix is in row-echelon form, write the corresponding system of equations. Finally, use substitution to find the remaining variables, and write the solution as an ordered triple. Always check that the solution makes the original equations true.

Practice: Solving a System of Three Linear Equations Using a Matrix

Identifying an Inconsistent System Using a Matrix

Apply the matrix method to determine if a system of linear equations is inconsistent, such as: $$ \left\{\begin{array}{l} x + y + 3z = 0 \\ x + 3y + 5z = 0 \\ 2x + 4z = 1 \end{array}\right. $$ First, write the augmented matrix for the equations. Then, use row operations to systematically eliminate variables, working towards row-echelon form. If during this process a row emerges with all zeros on the left side of the vertical line and a non-zero number on the right, it translates to a mathematically false statement like $$0 = 1$$. Because this results in a false statement, it indicates the system is inconsistent and has no solution.

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