Problem 14 - Algebra and Trigonometry

Question

Perform each matrix row operation and write the new matrix. $$ \left[\begin{array}{rrr|r} {3} & {-12} & {6} & {9} \\ {1} & {-4} & {4} & {0} \\ {2} & {0} & {7} & {4} \end{array}\right]^{\frac{1}{3} R_{1}} $$

Step-by-Step Solution

Answer

The new matrix that results from performing the row operation \(\frac{1}{3} R_{1}\) on the given matrix is \(\left[\begin{array}{rrr|r} {1} & {-4} & {2} & {3} \ {1} & {-4} & {4} & {0} \ {2} & {0} & {7} & {4} \end{array}\right]\)

Step 1: Identify the Matrix and the Operation

The matrix given in the question is a 3x4 matrix, and the operation to be done on this matrix is \(\frac{1}{3} R_{1}\), which means to divide every entry in the first row by 3.

Step 2: Apply the Operation to Row 1

The first row of the matrix is {3, -12, 6, 9}. Divide every entry by 3, resulting in {1, -4, 2, 3}.

Step 3: Update the Matrix

Replace the first row of the original matrix with the new values from the operation. The updated matrix is \(\left[\begin{array}{rrr|r} {1} & {-4} & {2} & {3} \ {1} & {-4} & {4} & {0} \ {2} & {0} & {7} & {4} \end{array}\right]\)

Key Concepts

3x4 matrix

A matrix is a rectangular array of numbers arranged in rows and columns. In linear algebra, matrices are used extensively for storing and manipulating data. The matrix described here is a '3x4 matrix', meaning it has 3 rows and 4 columns. Each element of a matrix can be identified by its position, denoted by two indices: the row number and the column number.
Understanding the dimensions of a matrix is crucial because it affects the operations you can perform. A 3x4 matrix has entries like \( a_{ij} \), where \( i \) represents the row and \( j \) represents the column. For example, in this matrix:

• The element at the first row and second column is \( -12 \).
• The element at the third row and fourth column is \( 4 \).

Knowing the layout of a 3x4 matrix sets the stage for effectively applying transformations and operations.

matrix transformation

Matrix transformation involves changing a matrix through various operations. Transformations can include scaling, rotation, or even changing the matrix structure through methods like adding or subtracting rows. They are fundamental operations in linear algebra and are used to solve systems of equations, among other applications.
In our example, we perform a specific transformation, known as row scaling, on the 3x4 matrix. Transformation does not alter the overall structure or dimensions of the matrix; it primarily changes the values within the matrix. Whether you are scaling a row, swapping rows, or applying more complex transformations, understanding these operations is essential for solving linear algebra problems effectively.

row scaling

Row scaling is a specific type of matrix transformation where each element of a row is multiplied by a non-zero scalar. This operation helps in simplifying rows when solving systems of linear equations, particularly when aiming for reduced row-echelon form.
In this exercise, we apply the row scaling operation \( \frac{1}{3} R_1 \), which means multiplying every element of the first row by \( \frac{1}{3} \).
Steps involved in row scaling are simple yet powerful:

• Select the row to be scaled.
• Multiply each element of this row by the given scalar (in this case, \( \frac{1}{3} \)).
• Update the matrix with these new values, leaving other rows unchanged.

This specific transformation aids in making computations easier while keeping the matrix equivalent in terms of systems it represents.

linear algebra methods

Linear algebra encompasses various methods and techniques for manipulating matrices and solving systems of equations. Several key methods include:

• Gaussian elimination - used for solving systems of linear equations.
• Matrix factorizations such as LU decomposition - which help simplify complex calculations.
• Determinant computation - to understand matrix properties like invertibility.

Row operations like scaling, swapping, and adding are foundational in these methods. For example, matrix row operations are vital while performing Gaussian elimination to solve for unknowns in a system.
By mastering these techniques, you can efficiently handle complex problems in engineering, physics, computer science, and other disciplines. Each method provides a different lens through which the structure and solutions of linear systems can be understood and solved.