Problem 13 - Algebra and Trigonometry

Question

Perform each matrix row operation and write the new matrix. $$ \left[\begin{array}{rrr|r} {2} & {-6} & {4} & {10} \\ {1} & {5} & {-5} & {0} \\ {3} & {0} & {4} & {7} \end{array}\right]^{\frac{1}{2} R_{1}} $$

Step-by-Step Solution

Answer

The new matrix after performing half of row 1 operation is \n\n\[ \left[\begin{array}{rrr|r} {1} & {-3} & {2} & {5} \\ {1} & {5} & {-5} & {0} \\ {3} & {0} & {4} & {7} \end{array}\right] \]

Step 1: Identify the operation.

The operation is identified as \(\frac{1}{2} R_{1}\), which means divide every element in the first row (row 1) by 2.

Step 2: Perform the operation.

Divide each element in the first row of the matrix by 2. This gives \n\n\[ \left[\begin{array}{rrr|r} {1} & {-3} & {2} & {5} \\ {1} & {5} & {-5} & {0} \\ {3} & {0} & {4} & {7} \end{array}\right] \]

Key Concepts

Matrix Algebra

Matrix algebra is a significant topic in advanced mathematics, extensively used in the fields ranging from engineering and physics to economics and social sciences.

At its core, matrix algebra is about performing algebraic operations on matrices which are rectangular arrays of numbers, symbols, or expressions organized in rows and columns. These operations include addition, subtraction, and multiplication of matrices, as well as scalar multiplication, where each element of a matrix is multiplied by a single number (scalar).

For example, in the exercise provided, the operation \(\frac{1}{2} R_{1}\) involves scalar multiplication of row 1. Such a maneuver is foundational in manipulating matrices, whether for solving systems of equations, transforming geometric figures, or even in complex tasks such as finding eigenvalues and eigenvectors. Understanding how to correctly perform these operations not only aids in solving matrix-related problems but also builds a foundation for further study in linear algebra.

Elementary Row Operations

Elementary row operations are tools that help us manipulate a matrix to achieve a desired form, and there are three types that any student of linear algebra should become familiar with:

• Type 1: Swap the positions of two rows.
• Type 2: Multiply a row by a nonzero scalar.
• Type 3: Add or subtract the multiple of one row to another row.

These operations are fundamental because they don't change the solutions when a matrix represents a system of linear equations, which is crucial for techniques like Gaussian elimination.

Looking at our exercise, when the operation \(\frac{1}{2} R_{1}\) is applied, it’s a Type 2 row operation. The significance of understanding these operations lies in their utility for simplifying matrices, which in turn simplifies the process of finding solutions. Through these fundamentals, one can systematically approach more complicated problems involving matrices.

Systems of Linear Equations

A system of linear equations is a collection of one or more linear equations involving the same set of variables. For example, \(2x - 6y + 4z = 10\) could be one equation in a system. Solving these systems is one of the main applications of matrix algebra.

To solve them, we often use a matrix to represent the system, and then perform row operations to reach what is called row-echelon form or reduced row-echelon form. This form makes it easier to see the solutions or to apply back-substitution to find them. Moreover, the augmented matrix, as seen in the exercise with the vertical line separating the coefficients of the variables from the constants, is especially useful in this context.

Understanding how to employ elementary row operations effectively is key in finding the solutions to these systems. When students grasp these concepts and apply them correctly, this not only aids in solving textbook exercises but also enriches their analytical skills in dealing with real-world problems that can be expressed as systems of linear equations.