Problem 19 - Intermediate Algebra

Question

Perform each of the following elementary row operations on the augumented matrix \(\left[\begin{array}{rrr}-3 & 1 & -6 \\ 1 & -4 & 4\end{array}\right].\) $$ -\frac{1}{3} R_{1} $$

Step-by-Step Solution

Answer

The new augmented matrix is \[\begin{array}{rrr} 1 & -\frac{1}{3} & 2 \\ 1 & -4 & 4 \end{array}\].

Step 1: Understand the Augmented Matrix

An augmented matrix represents a system of linear equations. In this case, the matrix is \[\begin{array}{rrr}-3 & 1 & -6 \1 & -4 & 4\end{array}\] which corresponds to two equations in the form \(-3x + y = -6\) and \(x - 4y = 4\).

Step 2: Identify the Elementary Row Operation

The operation given is \(-\frac{1}{3} R_1\). This means we will multiply each entry in the first row by \(-\frac{1}{3}\).

Step 3: Perform the Operation on Row 1

Apply the operation \(-\frac{1}{3} R_1\):- Multiply the first entry of row 1: \(-\frac{1}{3} \times -3 = 1\).- Multiply the second entry of row 1: \(-\frac{1}{3} \times 1 = -\frac{1}{3}\).- Multiply the third entry of row 1: \(-\frac{1}{3} \times -6 = 2\).

Step 4: Write the New Matrix

After applying the operation, the first row is replaced by the new row, while the second row remains unchanged. The new augmented matrix is:\[\begin{array}{rrr}1 & -\frac{1}{3} & 2 \1 & -4 & 4\end{array}\]

Key Concepts

Augmented Matrix

An augmented matrix is a compact and efficient way to represent a system of linear equations. It combines all the coefficients of the variables and the constants from the equations into a single matrix. This is done by aligning the matrices side-by-side: the coefficient matrix on one side, followed by a dividing line, and then the constants as the last column.

• For example, consider the linear equations: \(-3x + y = -6\) and \(x - 4y = 4\).
• The augmented matrix representing these equations is: \[\begin{array}{rrr}-3 & 1 & | & -6 \ 1 & -4 & | & 4 \end{array}\]

This form is particularly useful in linear algebra because it allows us to utilize matrix operations to solve systems of equations. By performing row operations and converting the matrix to an equivalent form, we can simplify solving the system.

System of Linear Equations

A system of linear equations refers to a collection of two or more linear equations that involve the same set of variables. The solution to a system of linear equations is the set of values for the variables that satisfy all the equations simultaneously.

• For instance, the system given by the equations \(-3x + y = -6\) and \(x -4y = 4\) involves the variables \(x\) and \(y\).
• To find the solution, we aim for values of \(x\) and \(y\) that make both equations true at the same time.

Systems of equations can have a unique solution, infinitely many solutions, or no solution at all. The augmented matrix is a tool that helps in restructuring the system to easily find such solutions using methods like Gaussian elimination.

Matrix Transformation

Matrix transformation involves performing operations on a matrix to convert it into a different form. This is particularly useful in solving systems of linear equations. One common method involves using elementary row operations to simplify the matrix.

Elementary Row Operations

These include:

• Row swapping: Exchanging two rows in the matrix.
• Row multiplication: Multiplying all elements of a row by a non-zero constant.
• Row addition: Adding a multiple of one row to another row.

In the context of the provided exercise, the transformation applied \(-\frac{1}{3} R_{1}\) involves multiplying every element of the first row of the matrix by \(-\frac{1}{3}\). After performing these operations, the matrix is altered and typically simplified, moving us closer to finding the solution of the system of equations. Matrix transformations like this make complex systems easier to handle by systematically reducing the equations.