Problem 22 - Intermediate Algebra

Question

Perform each of the following elementary row operations on the augmented matrix \(\left[\begin{array}{rrrr}3 & 6 & -9 & 0 \\ 1 & 5 & -2 & 1 \\ -2 & 2 & -2 & 5\end{array}\right]\) $$ -\frac{1}{2} R_{3} $$

Step-by-Step Solution

Answer

The updated third row is \([1, -1, 1, -\frac{5}{2}]\).

Step 1: Identify the Matrix

The given augmented matrix is \[\begin{bmatrix} 3 & 6 & -9 & 0 \ 1 & 5 & -2 & 1 \ -2 & 2 & -2 & 5\end{bmatrix} \].It has three rows and four columns. The task is to apply the operation to the third row.

Step 2: Understand the Operation

The operation is \(-\frac{1}{2} R_{3}\). This means each element of the third row \([-2, 2, -2, 5]\) of the matrix will be multiplied by \(-\frac{1}{2}\).

Step 3: Apply the Operation

Apply the operation \(-\frac{1}{2}\) to each element in the third row:1. \(-2 \times -\frac{1}{2} = 1\)2. \(2 \times -\frac{1}{2} = -1\)3. \(-2 \times -\frac{1}{2} = 1\)4. \(5 \times -\frac{1}{2} = -\frac{5}{2}\)So, the new third row becomes \([1, -1, 1, -\frac{5}{2}]\).

Step 4: Update the Matrix

Replace the third row of the original matrix with the new third row obtained:\[\begin{bmatrix}3 & 6 & -9 & 0 \1 & 5 & -2 & 1 \1 & -1 & 1 & -\frac{5}{2}\end{bmatrix}\]This is the updated augmented matrix after performing the operation.

Key Concepts

Augmented Matrix

An augmented matrix is a powerful tool commonly used in solving systems of linear equations. It combines the coefficient matrix and the constants from the right-hand side of the equations into one coherent matrix. This setup allows for efficient manipulation and application of elementary row operations.
It is structured in a way that each row of the matrix corresponds to an equation in the system. Meanwhile, each column before the vertical line represents coefficients of a variable, and the column after represents the results of the equation. For example, in the matrix:

• First row: [3, 6, -9 | 0]
• Second row: [1, 5, -2 | 1]
• Third row: [-2, 2, -2 | 5]

This corresponds to the equations:

• 3x + 6y - 9z = 0
• x + 5y - 2z = 1
• -2x + 2y - 2z = 5

By working with augmented matrices, you can systematically solve for the variables x, y, and z through transformations such as row operations.

Matrix Transformation

Matrix transformation involves operations that alter the structure of a matrix, aimed at simplifying the system represented by the matrix. Elementary transformations include row swapping, row multiplication by a scalar, and row addition or subtraction.
In the context of solving systems of equations, these transformations help in achieving an equivalent matrix that is easier to interpret and solve. For instance, multiplying a row by a non-zero scalar involves changing each element of the row by a consistent factor. This operation allows flexibility in transforming the matrix without changing the intrinsic properties of the equation system it represents.
In our example, applying the operation \(-\frac{1}{2} R_{3}\) means scaling the third row by \(-\frac{1}{2}\). Transformations like these are foundational in converting matrices to forms where solutions are readily obtainable, such as reduced row-echelon form.

Row Reduction

Row reduction is a method used to simplify augmented matrices for clearer, more straightforward solutions to systems of linear equations. It is sometimes called Gauss-Jordan elimination. By performing a series of specified row operations, row reduction transforms the matrix as close as possible to the identity matrix.
Key steps involved in row reduction include:

• Identifying pivot elements, usually chosen from the leftmost columns first.
• Performing row operations to create zeros in the column below the pivot.
• Proceeding to subsequent columns and rows until the matrix reaches its simplest form.

Using our example, after applying \(-\frac{1}{2} R_{3}\), the matrix steps closer to a form where further reduction techniques can help isolate variables and solve the system. The goal is ultimately to have a diagonal line of leading ones, simplifying the back-substitution process required to find variable values.

Intermediate Algebra

Intermediate algebra encompasses the algebraic techniques required to manipulate equations and expressions involving multiple variables. Understanding the concepts of matrices and their operations is essentially part of an intermediate algebra curriculum. Understanding these mathematical principles not only helps in solving equations but also strengthens comprehension of how these elements interact and transform through algebraic operations. For instance, grasping how to perform elementary row operations requires knowledge of scalar multiplication and the behavior of linear equations.
By building expertise in intermediate algebra concepts, students can adeptly handle augmented matrices, perform row reductions, and ultimately solve complex systems of equations. This foundational knowledge is not only crucial for algebraic problem-solving but also forms the cornerstone of more advanced mathematical fields.