Problem 40 - Precalculus

Question

Perform the indicated row operations on each augmented matrix. $$\left[\begin{array}{rrrr|r} 1 & 0 & -1 & 5 & 2 \\ 0 & 1 & 2 & 3 & -5 \\ 0 & 0 & 1 & -2 & 2 \\ 0 & 0 & 0 & 1 & 1 \end{array}\right] \begin{array}{l} R_{3}+2 R_{4} \rightarrow R_{3} \\ R_{2}-3 R_{4} \rightarrow R_{2} \\ R_{1}-5 R_{4} \rightarrow R_{1} \end{array}$$

Step-by-Step Solution

Answer

The final matrix is \( \begin{bmatrix} 1 & 0 & -1 & 0 & -3 \\ 0 & 1 & 2 & 0 & -8 \\ 0 & 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 1 & 1 \end{bmatrix} \).

Step 1: Perform Operation on Row 3

The operation is \( R_{3} + 2R_{4} \rightarrow R_{3} \). This means that we replace the third row, \( R_3 \), with the sum of itself and two times the fourth row, \( R_4 \). We compute each element of the new \( R_3 \):\[R_3 = [0, 0, 1, -2, 2] + 2 imes [0, 0, 0, 1, 1] = [0, 0, 1, -2+2, 2+2]= [0, 0, 1, 0, 4]\]The new matrix becomes:\[\begin{bmatrix}1 & 0 & -1 & 5 & 2 \0 & 1 & 2 & 3 & -5 \0 & 0 & 1 & 0 & 4 \0 & 0 & 0 & 1 & 1\end{bmatrix}\]

Step 2: Perform Operation on Row 2

Perform the operation \( R_{2} - 3R_{4} \rightarrow R_{2} \). This means replacing the second row, \( R_2 \), with the result of itself minus three times the fourth row, \( R_4 \). We compute:\[R_2 = [0, 1, 2, 3, -5] - 3 imes [0, 0, 0, 1, 1] = [0, 1, 2, 3-3, -5-3]= [0, 1, 2, 0, -8]\]Now, the matrix becomes:\[\begin{bmatrix}1 & 0 & -1 & 5 & 2 \0 & 1 & 2 & 0 & -8 \0 & 0 & 1 & 0 & 4 \0 & 0 & 0 & 1 & 1\end{bmatrix}\]

Step 3: Perform Operation on Row 1

Finally, execute the operation \( R_{1} - 5R_{4} \rightarrow R_{1} \). This involves replacing the first row, \( R_1 \), by subtracting five times the fourth row, \( R_4 \). Compute this as follows:\[R_1 = [1, 0, -1, 5, 2] - 5 imes [0, 0, 0, 1, 1] = [1, 0, -1, 5-5, 2-5]= [1, 0, -1, 0, -3]\]The final matrix is:\[\begin{bmatrix}1 & 0 & -1 & 0 & -3 \0 & 1 & 2 & 0 & -8 \0 & 0 & 1 & 0 & 4 \0 & 0 & 0 & 1 & 1\end{bmatrix}\]

Key Concepts

Augmented Matrix

An augmented matrix is an essential tool in linear algebra, especially when dealing with systems of linear equations. It essentially combines two matrices: the coefficient matrix and the constant matrix.
For the system of equations, the augmented matrix format allows you to align coefficients neatly alongside their corresponding constants.
This makes it much easier for analyzing and applying matrix operations.
The advantage of using an augmented matrix lies in its ability to streamline the process of solving systems of equations, by representing them in a compact form.
It provides a convenient and structured way to perform row operations, which are necessary for methods like Gaussian Elimination.
If you're dealing with multiple equations and variables, augmented matrices facilitate the organization and computation of solutions without getting bogged down with individual algebraic manipulations.

Elementary Row Operations

Elementary row operations are the backbone of matrix manipulation techniques used to simplify matrices or solve linear systems. These operations help transform matrices into easier forms, such as into row-echelon or reduced row-echelon form, making it simpler to find solutions or further analyze the system.
There are three types of row operations:

• Row switching: Swapping two rows if needed to create a more favorable matrix setup.
• Row multiplication: Multiplying all elements of a row by a non-zero scalar to adjust the equation.
• Row addition: Adding or subtracting multiples of rows to simplify or eliminate variables.

In our step-by-step solution, operations like adding multiples of one row to another are performed to modify the augmented matrix strategically.
These operations are fundamental in implementing Gaussian elimination and other matrix-solving techniques.

Gaussian Elimination

Gaussian Elimination is a systematic process to solve systems of linear equations, transforming matrices into a form from which solutions can be easily extracted. It's a widely used method in both theoretical and applied aspects of linear algebra.
The process is executed in several steps:

• Convert the augmented matrix into an upper triangular form using elementary row operations.
• Resolve any upper triangular matrix into a diagonal matrix, if necessary, entering a phase often referred to as back substitution.

Gaussian Elimination reduces down to transforming the matrix so that the lower left-hand corner of the matrix becomes zero-filled. This simplifies the system significantly.
Upon reaching row-echelon form, solutions for variables can be accessed easily through back substitution.
Understanding and applying Gaussian Elimination can be simplified by focusing on sequences of row operations that strategically pivot the matrix toward an increasingly simpler form.
This is not only efficient but also reveals the nature of the solution, whether it is unique, infinite, or if no solution exists.