Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Pivot the system about the circled element. \(\left[\begin{array}{cc|c}(2) & 4 & 8 \\ 3 & 1 & 2\end{array}\right]\)

Step-by-Step Solution

Answer

The short answer for pivoting the system about the circled element is: 1. Divide the first row by the circled element (2): \(R_1 = \frac{1}{2} R_1 \) Result: \[\left[\begin{array}{cc|c}1 & 2 & 4 \\\ 3 & 1 & 2\end{array}\right]\] 2. Make zeros below the pivot element: \(R_2 = R_2 - 3R_1\) Result: \[\left[\begin{array}{cc|c}1 & 2 & 4 \\\ 0 & -5 & -10\end{array}\right]\]

Step 1. Divide the first row

To get a 1 in the circled position, divide the entire first row by the circled element (2): \(R_1 = \frac{1}{2} R_1 \) This will result in the following matrix: \(\left[\begin{array}{cc|c}1 & 2 & 4 \\\ 3 & 1 & 2\end{array}\right]\)

Step 2. Make zeros below the pivot element

Now that we have a 1 in the circled position, we need to make zeros below it. To do this, we perform the following row operation: \(R_2 = R_2 - 3R_1\) This will result in the following matrix: \(\left[\begin{array}{cc|c}1 & 2 & 4 \\\ 0 & -5 & -10\end{array}\right]\) Now, the system is pivoted about the circled element as required.

Key Concepts

Row Operations

Row operations are the maneuvers we can perform on the rows of a matrix without changing the solution set of the corresponding system of linear equations. Commonly, we use three types of row operations in linear algebra:

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

Using these operations allows us to simplify or rearrange matrices into a more 'friendly' form, especially when attempting to solve a system of linear equations. They are powerful tools because, if used correctly, they maintain the essential properties and relationships between the original equations encoded within the matrix.

Elementary Row Operations

Elementary row operations hold the key to methods like Gaussian elimination. They are the basic moves we can make to transform a matrix while keeping the original system's solution intact. Each operation is an elementary action that helps us achieve a simpler form to read off solutions or to pivot around key elements in the matrix.

• To scale a row, multiply it by a non-zero number.
• To replace a row, add or subtract another row times a scalar.
• To interchange rows, simply swap their positions.

By applying these operations systematically, we can manipulate and pivot a matrix to a point where the solution to the system becomes apparent. These operations are reversible, and applying them in the reverse order will yield the original matrix.

Gaussian Elimination

Gaussian elimination is a method for solving systems of linear equations. It uses a sequence of elementary row operations to transform the matrix representing the system into a unique form, known as Row Echelon Form (REF) or sometimes into Reduced Row Echelon Form (RREF).
The goal is to create a diagonal of leading ones (pivots) and zeros everywhere below them:

• Begin with the leftmost nonzero column and create a pivot by making all entries below a selected one (the pivot) into zeros using row operations.
• Move to the next column and repeat the process for each subsequent pivot, working your way from left to right across the matrix.
• Once in REF or RREF, solve the equations represented by the matrix by back-substitution starting from the bottom row.

Gaussian elimination simplifies what might otherwise be a complicated system to a point where the solutions can easily be extracted.

System of Linear Equations

A system of linear equations consists of two or more equations that share a common set of variables. Mathematically, we seek a set of numbers that simultaneously make all the equations true. Representing this system using matrices is especially powerful because it allows us to apply procedural techniques such as Gaussian elimination to find solutions.

• In matrix terms, each equation becomes a row, and each column corresponds to a variable.
• The coefficients of the variables from the equations form the entries of the matrix.
• The solutions to the system, if they exist, are the values of the variables that satisfy all equations simultaneously.

Matrices enable a compact and systematic approach to solving these systems, often allowing us to identify whether a unique solution exists, if there are infinitely many solutions, or if the system is inconsistent with no solution.