Problem 23 - Precalculus: Mathematics for Calculus

Question

Perform the indicated elementary row operation. \(\left[\begin{array}{rrrr}2 & 1 & -3 & 5 \\ 2 & 3 & 1 & 13 \\ 6 & -5 & -1 & 7\end{array}\right]\) Add \(-3\) times Row 1 to Row 3

Step-by-Step Solution

Answer

The new Row 3 is \([0, -8, 8, -8]\).

Step 1: Identify the Rows and Operation

We need to perform an elementary row operation on the matrix. The matrix is a 3x4 matrix, with three rows and four columns. The operation required is to add (-3) times Row 1 to Row 3.

Step 2: Multiply Row 1 by -3

First, we will multiply each element in Row 1 by -3 to see what we add to Row 3. Row 1 is \([2, 1, -3, 5]\), and multiplying by -3 gives us \([-6, -3, 9, -15]\).

Step 3: Add the Result to Row 3

Next, we add the result from the previous step to Row 3. Row 3 is originally \([6, -5, -1, 7]\). Adding \([-6, -3, 9, -15]\) to it, we get: \([6-6, -5-3, -1+9, 7-15]\), which simplifies to \([0, -8, 8, -8]\).

Step 4: Write the New Matrix

Write out the new matrix with the updated Row 3. The new matrix is: \[\begin{bmatrix}2 & 1 & -3 & 5 \2 & 3 & 1 & 13 \0 & -8 & 8 & -8\end{bmatrix}\]

Key Concepts

Matrix Operations

Matrix operations are fundamental in solving linear equations, transforming geometric data, and dealing with complex systems in mathematics. At its core, these operations include addition, subtraction, and multiplication of matrices. Each operation follows specific rules that keep the structure of rows and columns intact.
When working with a matrix like the one given in the exercise, it is important to know how to perform operations such as row addition and scalar multiplication. These operations allow us to manipulate the matrix in ways that simplify our solutions to equations or transform our data.
In this exercise, an elementary row operation is performed, involving the matrix's rows. This type of manipulation is essential for tasks in algebra, as it maintains the relationships between the matrix's elements while changing its structure. Understanding these basic operations lays a strong foundation for more complex concepts in linear algebra.

Row Addition

Row addition is a type of elementary row operation used in matrix manipulation. It involves taking one row from a matrix, potentially multiplying it by a scalar, and adding it to another row. This can help simplify matrices and solve systems of linear equations.
In the problem, to "add \(-3\) times Row 1 to Row 3" means we first scale Row 1. The multiplication by \(-3\) yields \([-6, -3, 9, -15]\). Then, we perform the addition to Row 3, which alters its original form. This results in a new Row 3: \[ [6 - 6, -5 - 3, -1 + 9, 7 - 15] = [0, -8, 8, -8] \].
This process is critical when solving systems of equations using methods like Gaussian elimination. It allows for row transformation without changing the solution to the system. Row addition helps achieve row-echelon form, which is a reduced form where the matrix is easier to interpret.

Precalculus

Precalculus serves as a bridge between the concepts of algebra and the more complex topics found in calculus and higher-level mathematics. It helps build the mathematical foundation necessary for understanding calculus concepts. One of the areas covered is matrices and their applications, which are often introduced in a precalculus course.
Matrices can represent and solve systems of linear equations, and operations such as row addition are part of the techniques students learn to manipulate these matrices effectively. In a precalculus setting, being comfortable with these fundamental operations can make the transition to calculus smoother.
Understanding matrix operations also impacts other mathematical and real-world applications, including physics, engineering, and computer science. This makes precalculus an important stepping stone in the mathematical education journey.