Problem 38 - Precalculus

Question

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

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{bmatrix} 1 & 0 & 0 & 0 & | & 1 \\ 0 & 1 & 0 & 0 & | & -2 \\ 0 & 0 & 1 & 0 & | & 0 \\ 0 & 0 & 0 & 1 & | & -3 \end{bmatrix} \).

Step 1: Identify the Matrix and Row Operations

The given matrix is an augmented matrix, denoted as \[\begin{bmatrix} 1 & 0 & 4 & 0 & | & 1 \ 0 & 1 & 2 & 0 & | & -2 \ 0 & 0 & 1 & 0 & | & 0 \ 0 & 0 & 0 & 1 & | & -3 \end{bmatrix}\] We need to perform the following row operations:- Replace Row 2 \( (R_2) \) with \( R_2 - 2R_3 \).- Replace Row 1 \( (R_1) \) with \( R_1 - 4R_3 \).

Step 2: Perform the Operation on Row 2

Calculate the new Row 2 by performing the operation \( R_2 = R_2 - 2R_3 \):- Original Row 2: \([0, 1, 2, 0, | -2] \)- Row 3 (multiplied by 2): \([0, 0, 2, 0, | 0] \)- Subtract Row 3 from Row 2: \[ \begin{array}{ccccc} 0 & 1 & 2 & 0 & -2 \ - & 0 & 0 & -2 & 0 & 0 \hline 0 & 1 & 0 & 0 & -2 \end{array} \]The new Row 2 is \([0, 1, 0, 0, | -2] \). Update the matrix with this new row.

Step 3: Perform the Operation on Row 1

Calculate the new Row 1 by performing the operation \( R_1 = R_1 - 4R_3 \):- Original Row 1: \([1, 0, 4, 0, | 1] \)- Row 3 (multiplied by 4): \([0, 0, 4, 0, | 0] \)- Subtract Row 3 from Row 1: \[ \begin{array}{ccccc} 1 & 0 & 4 & 0 & 1 \ - & 0 & 0 & 4 & 0 & 0 \hline 1 & 0 & 0 & 0 & 1 \end{array} \]The new Row 1 is \([1, 0, 0, 0, | 1] \). Update the matrix with this new row.

Step 4: Write the Resulting Matrix

After performing the row operations, the updated augmented matrix becomes:\[\begin{bmatrix} 1 & 0 & 0 & 0 & | & 1 \0 & 1 & 0 & 0 & | & -2 \0 & 0 & 1 & 0 & | & 0 \0 & 0 & 0 & 1 & | & -3 \end{bmatrix}\]

Key Concepts

Understanding Augmented Matrices

Let's break down what an augmented matrix is. When dealing with systems of linear equations, an augmented matrix is a useful tool. It includes two parts:

• The coefficient matrix, which contains the coefficients of the variables.
• The constants matrix, which is derived from the equations' right side.

In our example, the augmented matrix is a 4x5 matrix. The first four columns correspond to the coefficients of four variables, while the last column contains the constants. Viewing them together allows us to handle multiple equations efficiently as one matrix, simplifying our operations.
Remember, the vertical line separating the data in an augmented matrix identifies which numbers are coefficients and which are constants.
Augmented matrices are fundamental when learning about row operations, making it easier to organize and systematically solve systems of equations.

The Role of Matrix Transformation

Matrix transformation refers to the process of manipulating a matrix to achieve a desired form. In the context of solving equations, these transformations help simplify matrices through operations that retain equivalency.
When you perform operations like swapping rows, scaling them, or adding multiples of rows, you are conducting matrix transformations. The goal is often to transform a matrix into its row-echelon or reduced row-echelon form, which makes it easier to interpret solutions or apply methods like Gaussian elimination.Consider the given matrix, which was transformed with specific row operations to achieve the following form:

• From \( R_2 = R_2 - 2R_3 \), you clarified dependency between these rows by eliminating certain coefficients.
• From \( R_1 = R_1 - 4R_3 \), you refined the top row to reduce unnecessary complexity.

Such transformations systematically simplify matrices and aid in isolating variables, making the problem-solving process more straightforward.

Elementary Row Operations Explained

Elementary row operations are the backbone of manipulating and solving linear systems using matrices. There are three primary types:

• Row Swapping: Switches the positions of two rows. Useful for organizing the matrix for easier transformations.
• Row Multiplication: Involves multiplying all elements of a row by a non-zero scalar. Allows for adjusting the scaling of equations.
• Row Addition/Subtraction: You can add or subtract the multiples of one row from another. It's crucial for eliminating variables and simplifying systems.

In the problem provided, the operations involve row subtraction which helps in achieving a simpler, equivalent matrix by eliminating terms.
By subtracting multiples of Row 3 from Row 1 and Row 2, unnecessary complexities were removed. This is known as "zeroing out" elements, a common technique when moving towards solutions.
These operations are reversible, preserving the solutions of the original equations, thus widely used for accuracy and effectiveness in mathematical computation.