Problem 34 - Precalculus

Question

Perform the indicated row operations on each augmented matrix. $$\left[\begin{array}{rrr|r} 1 & -2 & 1 & 3 \\ 0 & 1 & -2 & 6 \\ -3 & 0 & -1 & -5 \end{array}\right] \quad R_{3}+3 R_{1} \rightarrow R_{3}$$

Step-by-Step Solution

Answer

The updated matrix is \[ \begin{bmatrix} 1 & -2 & 1 & 3 \\ 0 & 1 & -2 & 6 \\ 0 & -6 & 2 & 4 \end{bmatrix} \].

Step 1: Understand the Row Operation

We are given the operation \( R_3 + 3R_1 \rightarrow R_3 \). This means we will replace the third row, \( R_3 \), with the sum of \( R_3 \) and three times the first row, \( R_1 \), of the matrix.

Step 2: Multiply the First Row

First, multiply each element of the first row, \( R_1 = [1, -2, 1, 3] \), by 3: \( 3R_1 = [3 \cdot 1, 3 \cdot (-2), 3 \cdot 1, 3 \cdot 3] = [3, -6, 3, 9] \).

Step 3: Add to the Third Row

Now, add the scaled row, \( 3R_1 = [3, -6, 3, 9] \), to the third row, \( R_3 = [-3, 0, -1, -5] \):- First column: \(-3 + 3 = 0\) - Second column: \(0 + (-6) = -6\)- Third column: \(-1 + 3 = 2\)- Fourth column: \(-5 + 9 = 4\)Thus, \( R_3 \) becomes \([0, -6, 2, 4]\).

Step 4: Write the New Matrix

After performing the row operation, the new augmented matrix is: \[\begin{bmatrix} 1 & -2 & 1 & 3 \ 0 & 1 & -2 & 6 \ 0 & -6 & 2 & 4 \end{bmatrix} \]

Key Concepts

Augmented Matrix

An augmented matrix is a way to represent a system of linear equations in matrix form, where the coefficients of the variables and the constants on the right-hand side of the equations are included in a single matrix. The vertical bar in the matrix separates the coefficients of the variables from the constants. For example, in our original matrix:\[\begin{bmatrix} \ 1 & -2 & 1 & | & 3 \ 0 & 1 & -2 & | & 6 \ -3 & 0 & -1 & | & -5 \ \end{bmatrix} \]- The left side before the vertical bar consists of the coefficients of the variables in the equations.- The right side represents the constants from each equation.This form is exceptionally useful in simplifying and solving systems of equations by utilizing matrix operations. The entire matrix is used to perform calculations such as elementary row operations that lead us to a solution for the equations represented.

Elementary Row Operations

Elementary row operations are fundamental tools in matrix algebra, allowing us to solve systems of equations. These operations consist of three main types:

• Row Switching: Interchanging two rows of a matrix.
• Row Multiplication: Multiplying all elements of a row by a nonzero scalar.
• Row Addition: Adding a multiple of one row to another row.

These operations do not change the solutions of the system of equations, making them ideal for transforming a matrix into a more manageable form, such as echelon or reduced row echelon form. For instance, the operation \( R_3 + 3R_1 \rightarrow R_3 \) used in the given exercise employs row addition. We took the first row, scaled it by 3, and added it to the third row, effectively transforming the matrix to simplify solving the corresponding system of equations.

Matrix Transformation

Matrix transformation involves using operations like elementary row operations to convert a matrix into a different form. This process is often used to solve linear equations or to find inverses of matrices. By systematically applying these operations, we can transform a matrix into what is known as row echelon form or reduced row echelon form.Transforming matrices through these operations helps in solving systems by:

• Eliminating variables by forming zeros in strategic positions.
• Making it easier to back-substitute and find solutions for variables.

In the original exercise, we used matrix transformation to alter the third row, using the operation \( R_3 + 3R_1 \rightarrow R_3 \), which removed coefficients of certain terms to simplify the equation. This step-by-step methodology enables us to clear up and isolate variables, gradually working our way to find the values that satisfy the given system of conditions.