Problem 20 - Advanced Engineering Mathematics

Question

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(x_{1}+2 x_{2}+\quad x_{4}=0\) \(4 x_{1}+9 x_{2}+x_{3}+12 x_{4}=0\) \(3 x_{1}+9 x_{2}+6 x_{3}+21 x_{4}=0\) \(x_{1}+3 x_{2}+x_{3}+9 x_{4}=0\)

Step-by-Step Solution

Answer

The system has infinitely many solutions, expressed as \(x_1 = 17t, x_2 = -9t, x_3 = 2t, x_4 = t\).

Step 1: Set up the Augmented Matrix

Start by translating the system of equations into an augmented matrix. For our system, the corresponding matrix is:\[\begin{bmatrix}1 & 2 & 0 & 1 & | & 0 \4 & 9 & 1 & 12 & | & 0 \3 & 9 & 6 & 21 & | & 0 \1 & 3 & 1 & 9 & | & 0 \end{bmatrix}\]

Step 2: Begin Row Reduction (Eliminate X1)

To eliminate the variables below \(x_1\), pivot on the first row and make other elements in column 1 to zero.Subtract 4 times Row 1 from Row 2, 3 times Row 1 from Row 3, and Row 1 from Row 4:\[\begin{bmatrix}1 & 2 & 0 & 1 & | & 0 \0 & 1 & 1 & 8 & | & 0 \0 & 3 & 6 & 18 & | & 0 \0 & 1 & 1 & 8 & | & 0\end{bmatrix}\]

Step 3: Make Additional Reductions (Eliminate X2)

Subtract Row 2 from Row 4 and Row 2 multiplied by 3 from Row 3 to make zeros in the second column below the pivot:\[\begin{bmatrix}1 & 2 & 0 & 1 & | & 0 \0 & 1 & 1 & 8 & | & 0 \0 & 0 & 3 & -6 & | & 0 \0 & 0 & 0 & 0 & | & 0\end{bmatrix}\]

Step 4: Analyze Row Echelon Form

Notice that the fourth row is all zeroes. This indicates a free variable can be present. We now have a system of three non-zero equations with four unknowns, meaning we have infinitely many solutions. Assign a parameter \(t\) to \(x_4\) and express other variables in terms of \(t\).

Step 5: Express Variables with a Parameter

From the third row, \(3x_3 - 6t = 0\) so \(x_3 = 2t\).From the second row, \(x_2 + x_3 + 8t = 0\) so \(x_2 = -9t\).Substitute into the first row: \(x_1 + 2(-9t) + t = 0\) yields \(x_1 = 17t\).

Step 6: Write the Solution

The solution in terms of the parameter \(t\) is:\[\begin{align*}x_1 &= 17t, \x_2 &= -9t, \x_3 &= 2t, \x_4 &= t\end{align*}\]Thus, the set of solutions is a line in four-dimensional space.

Key Concepts

Gauss-Jordan elimination

Gauss-Jordan elimination is a powerful method used to solve systems of linear equations by transforming the system's augmented matrix into reduced row-echelon form. This form is unique for a given system, making it easier to find the solutions.

The process involves performing a series of row operations to simplify the matrix:

• Swapping rows.
• Multiplying a row by a nonzero scalar.
• Adding or subtracting the product of a row and a scalar to another row.

Starting with the leftmost pivot position (the leading 1 in each row), the goal is to zero out all other entries in the column containing the pivot.

Once the matrix is in reduced row-echelon form, results can be easily read off, and if a solution exists, it is found straight away. Gauss-Jordan is preferred when you wish to find all solutions, free variables, or parametric solutions.

system of linear equations

A system of linear equations is a collection of one or more equations involving the same set of variables. The primary goal is to find the values of these variables that satisfy all equations simultaneously.

In our exercise, we have:

• Four equations.
• Variables: \(x_1, x_2, x_3, x_4\).

Linear systems can have:

• One solution (when lines intersect at a single point).
• No solutions (when lines are parallel and never intersect).
• Infinitely many solutions (when all lines coincide or overlap in certain dimensions).

For our system, since there are more variables than equations, we explore the possibility of infinitely many solutions by introducing parameters.

augmented matrix

An augmented matrix is a compact way to represent a system of equations. It includes both the coefficients of the variables and the constants from the equation's right side.

For the system in our exercise, the augmented matrix started as:\[\begin{bmatrix} 1 & 2 & 0 & 1 & | & 0 \ 4 & 9 & 1 & 12 & | & 0 \ 3 & 9 & 6 & 21 & | & 0 \ 1 & 3 & 1 & 9 & | & 0 \end{bmatrix}\]The vertical line separates the coefficients from the constants on the right side of the equations.

Using this matrix format simplifies computations and row operations, essential for processes like Gaussian and Gauss-Jordan elimination.

row reduction

Row reduction is an essential technique in linear algebra used to simplify matrices and solve linear systems. It involves performing row operations to bring a matrix closer to row-echelon or reduced row-echelon form.

There are three types of elementary row operations:

• Swap: Exchange two rows.
• Scale: Multiply a row by a nonzero constant.
• Add/Subtract: Replace one row with itself plus a multiple of another.

In our scenario, row reduction consisted of:

• Making lower elements in the pivot columns zero.
• Identifying and using leading entries to simplify the system.
• Resulting configuration showed third row dependency, indicating redundancy.

The result was a simplified matrix, revealing free variables and the solution's parametric form.