Problem 43 - Algebra and Trigonometry

Question

Exercises \(41-43\) will help you prepare for the material covered in the first section of the next chapter. Consider the following array of numbers: $$\left[\begin{array}{rrr} {1} & {2} & {-1} \\ {4} & {-3} & {-15} \end{array}\right]$$ Rewrite the array as follows: Multiply each number in the top row by -4 and add this product to the corresponding number in the bottom row. Do not change the numbers in the top row.

Step-by-Step Solution

Answer

The matrix after transformation is: \[ \left[ \begin{array}{ccc} 1 & 2 & -1 \ 0 & -11 & -11 \end{array} \right]\].

Step 1: Identifying the Matrix

The Array given is a 2x3 matrix: \[ \left[ \begin{array}{ccc} 1 & 2 & -1 \ 4 & -3 & -15 \end{array} \right] \]. There are two rows and three columns in this matrix.

Step 2: Performing Row Operations

Perform the operation specified: Multiply every element in the first row by -4 and add this to the corresponding element in the second row. This results in the following matrix: \[ \left[ \begin{array}{ccc} 1 * -4 + 4 & 2 * -4 + -3 & -1 * -4 + -15 \end{array} \right] \].

Step 3: Simplifying the Matrix

Simplify the matrix obtained in Step 2 by performing each operation: \[ \left[ \begin{array}{ccc} -4 + 4 & -8 + -3 & 4 - 15 \end{array} \right] = \left[ \begin{array}{ccc} 0 & -11 & -11 \end{array} \right] \].

Key Concepts

Understanding a 2x3 Matrix

Matrices are a fundamental concept in mathematics and closely related to linear algebra. A matrix is a two-dimensional array of numbers arranged in rows and columns. The terminology "2x3 matrix" is used to describe a specific type of matrix.

• The "2" signifies that there are 2 rows in the matrix.
• The "3" indicates that there are 3 columns.

In a 2x3 matrix like \[\begin{bmatrix} 1 & 2 & -1 \ 4 & -3 & -15 \end{bmatrix}\]the numbers are organized so that we can easily read or manipulate the data. Each entry in the matrix is known as an "element" and can be located using its row and column numbers. Understanding the dimensions and layout of matrices is crucial as it lays the foundation for performing various operations like addition, subtraction, and multiplication. It is also essential when working with matrix transformations and solving linear systems.

Executing Elementary Row Operations

Elementary row operations are tools used in linear algebra to manipulate the rows within a matrix. These operations allow us to perform procedures like making a matrix simpler or finding its inverse. In this context, we use these operations to transform a matrix systematically.

• Row addition: Add a multiple of one row to another row.
• Row exchange: Swap two rows.
• Row scaling: Multiply a row by a non-zero scalar.

For example, when instructed to multiply each element in the top row of the matrix by -4 and add it to the corresponding element in the bottom row, we are doing an instance of the row addition operation. This alters the second row without affecting the first row. The result of these operations can serve as an important step towards further simplifications or solving matrix equations.

Simplifying the Matrix

Once we perform the necessary elementary row operations on a matrix, the next step usually involves simplifying it further. Simplifying a matrix usually means making calculations easy and representing the matrix in a cleaner form.

In our example, after applying the row operation of adding \[-4 \times \begin{bmatrix} 1 & 2 & -1 \end{bmatrix}\] to the second row \[\begin{bmatrix} 4 & -3 & -15 \end{bmatrix} \], we end up with\[\begin{bmatrix} 0 & -11 & -11 \end{bmatrix} \].
This simplification shows all operations have been calculated, and the resulting row is zeroed out, except for the values which are derived directly from the calculations. Simplifying can sometimes lead to solving a system of equations or preparing the matrix for further transformations. Simpler matrices make it easier to solve or interpret mathematical models and provide insights into the properties of the matrix being analyzed.