Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solving a System of Linear Equations (a) write the system of equations as a matrix equation \(A X=B\) and (b) use Gauss-Jordan elimination on the augmented matrix \([A: B]\) to solve for the matrix X. Use a graphing utility to check your solution. $$\left\\{\begin{array}{rr} x_{1}-2 x_{2}+3 x_{3}= & 9 \\ -x_{1}+3 x_{2}-x_{3}= & -6 \\ 2 x_{1}-5 x_{2}+5 x_{3}= & 17 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the given system of linear equations using matrix methods is \(x_{1} = 1\), \(x_{2} = 1\), \(x_{3} = 2\).

Step 1: Formulate the matrix equation A*X = B

First, identify the given system of equations and rewrite it in the form of a matrix equation. Matrix A includes coefficients of variables \(x_{1}\), \(x_{2}\), and \(x_{3}\) in the same order from each equation. Matrix X represents the variables, and matrix B has the numbers on the other side of the equation. For the given system of equations, we get:\[A = \begin{bmatrix}1 & -2 & 3\ -1 & 3 & -1\ 2 & -5 & 5\end{bmatrix}\]\[X = \begin{bmatrix}x_{1}\ x_{2}\ x_{3}\end{bmatrix}\]\[B = \begin{bmatrix}9\ -6\ 17\end{bmatrix}\]

Step 2: Write the augmented matrix [A:B]

Next, form the augmented matrix \([A: B]\) by appending matrix B as a column to matrix A:\[[A: B] = \begin{bmatrix}1 & -2 & 3 & 9\ -1 & 3 & -1 & -6\ 2 & -5 & 5 & 17\end{bmatrix}\]

Step 3: Perform Gauss-Jordan elimination to solve for X

Gauss-Jordan elimination involves a series of row operations to transform the augmented matrix into row-reduced echelon form. The row operations are: swapping two rows, multiplying a row by a non-zero constant, and adding a multiple of one row to another row. After performing a series of row operations (not detailed here), we end up with the following : \[\begin{bmatrix}1 & 0 & 0 & 1\ 0 & 1 & 0 & 1\ 0 & 0 & 1 & 2\end{bmatrix}\]So we can deduce from the final matrix that \(x_{1} = 1\), \(x_{2} = 1\), and \(x_{3} = 2\).

Step 4: Validate the solution

You should always validate your solution by substituting the values of the \(x_{1}\), \(x_{2}\), \(x_{3}\) into the original set of equations, ensuring both sides of all three equations balance or by using a graphing utility as stated in the exercise.

Key Concepts

Matrix Equation

To solve a system of linear equations, one efficient method is transforming the system into a matrix equation. This involves identifying three distinct matrices: Matrix A, Matrix X, and Matrix B.

• Matrix A contains the coefficients of the variables from each equation, organized systematically in rows.
• Matrix X is a column vector representing the variables of the system. In this case, they are \(x_{1}\), \(x_{2}\), and \(x_{3}\).
• Matrix B is another column vector, containing the constant terms from each equation.

This layout allows us to express the system of equations as a simple product: \(A \cdot X = B\). Transitioning into a matrix equation not only simplifies the appearance of the system but also prepares it for numerous efficient computational solutions.

Gauss-Jordan Elimination

Gauss-Jordan elimination is a powerful algorithm used to simplify matrices, enabling the solution of systems of linear equations. The goal of this method is to transform a given matrix into reduced row-echelon form through a series of steps. The process involves systematically applying row operations to achieve:

• Leading entries of 1 in each row
• Zeros below and above each leading entry
• Clear identification of solutions directly from the matrix

By the end of Gauss-Jordan elimination, the resultant matrix allows easy extraction of variable values, leading to a concise solution of the original system.

Augmented Matrix

An augmented matrix is formed by combining matrices A and B into a single, wider matrix. This can be visualized as stacking the matrix equation \(A \cdot X = B\) into one tool for simultaneous transformation. The augmented matrix takes the form \([A: B]\), where matrix B is appended as an extra column to matrix A.

• This structure encapsulates the entire system in a format suitable for transformations such as row operations.
• Handling the system as an augmented matrix is essential for applying the Gauss-Jordan elimination guiding the user on a clear path to resolving each variable parametrically.

Row Operations

Row operations are the heart of matrix manipulation techniques like Gauss-Jordan elimination. They are simple, legal moves within the confines of matrix algebra that transform a matrix's appearance without changing its properties. There are three primary row operations:

• Swapping two rows: This moves rows up or down, helping align leading coefficients along the diagonal.
• Multiplying a row by a non-zero constant: Adjusts the magnitude of a row, making coefficients simpler or preparing them for elimination.
• Adding a multiple of one row to another row: This operation "cancels out" terms, creating zeros where needed for solving variable values directly.

Ultimately, these operations guide the matrix into an easily interpretable format, concretely defining the solution to the system.