Problem 39 - College Algebra Essentials

Question

Consider the following array of numbers:$$\left[\begin{array}{ccc}1 & 2 & -1 \\\ 4 & -3 &-15\end{array}\right]$$ 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 new array is \[\begin{array}{ccc}1 & 2 & -1 \0 & -11 & -11 \\end{array}\]

Step 1: Multiply

Firstly, multiply each number in the top row by \( -4 \). This results in the values \( -4, -8, 4 \).

Step 2: Sum with Corresponding Numbers

Next, add these computed values to the corresponding numbers in the bottom row. This means \( -4 + 4 = 0, -8 - 3 = -11, 4 -15 = -11 \).

Step 3: Rewrite Array

The final step is to write the new array. The top row remains \( 1, 2, -1 \) while the bottom row becomes the sums calculated in the previous step, which are \( 0, -11, -11 \). Thus the new array is: \[\begin{array}{ccc} 1 & 2 & -1 \ 0 & -11 & -11 \\end{array}\]

Key Concepts

Elementary Row Operations

Elementary row operations are fundamental techniques used in matrix operations to simplify matrices and solve systems of linear equations. They include a set of moves:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting a multiple of one row to another row.

These operations are essential in matrix algebra because they allow us to transform matrices into simpler forms, such as row-echelon form or reduced row-echelon form, to make calculations and understandings easier.
In the given exercise, multiplying each number in the top row by \( -4 \) and adding the resulting product to the corresponding number in the bottom row is an example of the third type of elementary row operation. It transforms the matrix into a simpler form without changing its underlying properties.

Matrices

Matrices are rectangular arrays of numbers, expressions, or symbols arranged in rows and columns. They play a critical role in various areas of mathematics and applied sciences. Matrices are primarily used to solve linear equations, perform transformations, and represent data.
The dimensions of a matrix are given by the number of rows and columns it has, described as \( m \) x \( n \), where \( m \) is the row count and \( n \) is the column count. In our exercise, we started with a \( 2 \times 3 \) matrix.
Matrices can be manipulated through a range of operations such as addition, subtraction, multiplication, and the application of elementary row operations, which we used in the exercise to simplify its form.

Matrix Transformations

Matrix transformations refer to operations that alter a matrix or the space the matrix represents. These transformations can change the matrix's elements, structure, or properties while retaining certain core characteristics.

• Transformations often aim to simplify matrices, making them easier to analyze or solve.
• They can provide insights into the solutions of systems represented by the matrix.

In the context of our exercise, the transformation involved using elementary row operations to change the bottom row of the matrix while keeping the top row intact. This transformation effectively adjusted the basic structure of the matrix but preserved the relationships between its numbers. By applying this operation, a new matrix is formed, providing a different perspective through which to view or analyze the data it represents. This approach is invaluable in fields where matrices are used to solve complex systems.