Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Pivot the system about the circled element. \(\left[\begin{array}{rrr|r}1 & 2 & 3 & 5 \\ 0 & -3 & 3 & 2 \\ 0 & 4 & -1 & 3\end{array}\right]\)

Step-by-Step Solution

Answer

The given matrix is already simplified with the pivot element as "1" in the first row, first column. No row operations are needed, and the matrix remains unchanged: \(\left[\begin{array}{rrr|r}1 & 2 & 3 & 5 \\ 0 & -3 & 3 & 2 \\ 0 & 4 & -1 & 3 \end{array}\right]\)

Step 1: Identify the Pivot Element

We assume that the pivot (leading) element is the "1" in the first row, first column.

Step 2: Eliminate Coefficients in the Pivot Column

The pivot element is in the first column and row. To eliminate coefficients in the pivot column, we'll perform row operations for the second and third rows. To make the second element in the first column 0, we don't have to do anything since it's already 0. To make the third element in the first column 0, we again won't have to perform any operations since it's already 0.

Step 3: Check the Result

As it turns out, the given matrix is already simplified since the coefficients in the first column have already been eliminated below the pivot element. The matrix remains unchanged: \(\left[\begin{array}{rrr|r}1 & 2 & 3 & 5 \\ 0 & -3 & 3 & 2 \\ 0 & 4 & -1 & 3 \end{array}\right]\)

Key Concepts

Pivot Element

In linear algebra, the concept of a pivot element is fundamental in the process of transforming matrices. A pivot element is the leading non-zero element in a row. It plays a crucial role when performing Gaussian elimination or other matrix simplification techniques. We select the pivot element to guide us in eliminating the coefficients in its column, thereby simplifying the system to a row-echelon or reduced row-echelon form.
Choosing the right pivot is crucial for the simplification process:

• It helps in systematically eliminating variables to isolate solutions.
• It should be a non-zero entry to make the elimination process effective.

In the given exercise, the pivot was the "1" in the first row and first column. This pivot allows us to work through the first column efficiently and prepare the matrix for further simplification operations if necessary.

Row Operations

Row operations are the heart of matrix manipulations. They are conducted to alter matrices without changing their inherent properties. There are three main types of row operations:

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

In the step-by-step solution, the operations needed were not extensive. Since the elements below the pivot in the first column were already zeros, no additional row operations were necessary to simplify the matrix further. Understanding when to apply these operations helps students simplify systems efficiently and correctly.

Matrix Simplification

Matrix simplification refers to the process of transforming a matrix into simpler forms like row-echelon or reduced row-echelon form. These forms make it easier to solve systems of linear equations. The simplified forms help us demonstrate uniqueness and existence of solutions:

• Row-echelon form (REF) demands each leading coefficient to be 1, and all elements below them in their column to be zero.
• Reduced row-echelon form (RREF) further reduces all elements in the pivot column above and below the leading 1 to zero.

In the provided matrix, since the necessary column was already simplified with zeros below the pivot, the task was straightforward. The matrix remains unchanged, ready to yield solutions without further simplification, highlighting an inherently efficient path towards solving the given system.