Problem 21 - Precalculus: Mathematics for Calculus

Question

Perform the indicated elementary row operation. \(\left[\begin{array}{rrrr}-1 & 1 & 2 & 0 \\ 3 & 1 & 1 & 4 \\ 1 & -2 & -1 & -1\end{array}\right]\) Add 3 times Row 1 to Row 2

Step-by-Step Solution

Answer

The new Row 2 is \(0, 4, 7, 4\).

Step 1: Identify Rows Involved

Identify the rows involved in the elementary row operation. We will use Row 1 and Row 2 from the matrix. Row 1 is \(-1, 1, 2, 0\) and Row 2 is \(3, 1, 1, 4\).

Step 2: Multiply Row 1

Multiply each element of Row 1 by 3. The operation yields \(3\times(-1) = -3\), \(3\times1 = 3\), \(3\times2 = 6\), \(3\times0 = 0\). The resulting row is \(-3, 3, 6, 0\).

Step 3: Perform Row Addition

Add the results from Step 2 to the corresponding elements in Row 2. Calculate each new element in Row 2: \(3 + (-3) = 0\), \(1 + 3 = 4\), \(1 + 6 = 7\), \(4 + 0 = 4\).The new Row 2 becomes \(0, 4, 7, 4\).

Step 4: Write the New Matrix

Replace the old Row 2 with the new Row 2 in the original matrix, resulting in the new matrix:\[\begin{array}{rrrr}-1 & 1 & 2 & 0 \0 & 4 & 7 & 4 \1 & -2 & -1 & -1\end{array}\]

Key Concepts

Matrix Operations

Matrix operations are key tools in linear algebra that allow us to manipulate matrices for various mathematical purposes. One fundamental operation is the elementary row operation, which includes actions such as row swapping, row multiplication, and row addition. In our specific exercise, the operation involves adding a multiple of one row to another, which is known as row addition. This particular technique is incredibly useful for solving linear equations and simplifying matrices. Let's break down why these operations are so significant:

• Flexibility: Matrix operations can transform matrices into simpler forms, aiding in problem-solving.
• Simplification: They help reduce complex systems into more manageable equations.
• Central Role: Operations like these are integral in computational algorithms and data transformations.

Understanding matrix operations gives students the power to manage and solve complex systems efficiently.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations, transform matrices, and find matrix inverses. It involves a sequence of operations on the rows of the matrix to achieve row-echelon form or even reduced row-echelon form. In our case, we've employed an elementary row operation, a key step in the Gaussian elimination process.

Here’s how Gaussian elimination works in simple terms:

• Pivot Element: This is chosen to eliminate variable coefficients below the pivot row.
• Row Operations: Similar to our exercise, this could mean adding, subtracting, or multiplying rows.
• Upper Triangular Form: The matrix is simplified so that all the numbers below the pivot rows become zeros.

By understanding Gaussian elimination, one can systematically reduce complex systems into simpler forms, ready to be solved through back-substitution. It's an essential technique not just in math but also in fields like engineering and computer science.

Systems of Linear Equations

Systems of linear equations are collections of one or more linear equations involving the same variables. Solving these systems is a central challenge in algebra, and techniques like matrix operations and Gaussian elimination are directly applied to them.

A system of linear equations can be represented in matrix form, where the system is expressed in terms of a matrix equation, often noted as \(Ax = b\). Here, \(A\) is the matrix containing the coefficients, \(x\) is the column vector of variables, and \(b\) represents the output values.

Key elements when dealing with systems are:

• Consistency: Determining if a solution exists for the given equations.
• Uniqueness: Whether the system has a single solution, multiple solutions, or none at all.
• Solution Methods: Methods such as substitution, elimination, and using matrices help in finding solutions.

By mastering systems of linear equations, students can solve various real-world problems that require finding unknowns based on known relationships between quantities.