Problem 39 - College Algebra

Question

Exercises 37-39 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 new array is: \[\left[\begin{array}{rrr}1 & 2 & -1 \ 0 & -11 & -11\end{array}\right]\]

Step 1: Understanding the Matrix

This matrix is in the form of a 2x3 matrix with two rows and three columns. Each position in this matrix is generally referred to by two index numbers that correspond to its row and column. The first number identifies its row and the second its column.

Step 2: Applying the Operations to the Matrix

The operation defined in the problem is to multiply each number in the top row of the matrix by -4 and add the result to the corresponding number in the bottom row. This doesn't affect the top row, so the top row remains [1, 2, -1]. Now multiply each top-row number by -4: [-4, -8, 4], and add these results to the corresponding numbers in the second row. For the first column, 4 plus -4 equals 0. For the second column, -3 plus -8 equals -11. And for the third column, -15 plus 4 equals -11.

Step 3: Forming the New Matrix

After applying the operation, the bottom row becomes [0, -11, -11]. Therefore, the new matrix is [1, 2, -1] for top row and [0, -11, -11] for the bottom row.

Key Concepts

Matrix Transformation

A matrix transformation involves the application of a set of operations that change the elements of a matrix in a systematic way. In this exercise, the transformation is achieved by altering the bottom row of a given 2x3 matrix. The top row is multiplied by each element by -4, and each resulting product is added to the respective element of the bottom row. This kind of transformation is often used to simplify matrices or solve systems of equations. It affects the matrix's numerical values but not its overall structure, which remains a 2x3 form. Such transformations are essential in linear algebra as they help facilitate many operations like determining matrix inverses or simplifying equations.

Row Operations

Row operations are fundamental techniques in matrix arithmetic. They include operations like swapping rows, multiplying a row by a constant, and adding multiples of one row to another. For this specific exercise, the operation is focused on the third type: adding a multiple of one row to another. By multiplying the top row by -4 and adding the result to the bottom row, we perform an operation that doesn't alter the first row but changes the bottom row. Such operations are used in methods like Gaussian elimination for solving linear equations or transforming matrices into row-echelon form. They are straightforward, yet powerful, making matrix manipulation much more manageable.

2x3 Matrix

A 2x3 matrix is a grid composed of numbers arranged in two rows and three columns. Each element within the matrix can be individually referenced by its position in terms of row and column. The matrix explained in this exercise initially takes the form: \[ \begin{bmatrix} 1 & 2 & -1 \ 4 & -3 & -15 \end{bmatrix} \] The numbers within this matrix can represent various real-world data, like coefficients in linear equations. The dimensions (2 rows and 3 columns) categorize it among non-square matrices, which are key to many operations in advanced mathematics. Despite not being square, transformations and row operations, like those in the exercise, can still be applied effectively.

Matrix Arithmetic

Matrix arithmetic involves operations such as addition, subtraction, multiplication, and scalar multiplication. In this specific exercise, the arithmetic is conducted through scalar multiplication combined with addition. Each element of the top row is multiplied by -4, then added to the corresponding element of the bottom row. Matrix arithmetic is fundamental because it allows for manipulation of data and solving linear systems, making transformations and computations more intuitive. While these operations seem simple, they are foundational to more complex techniques such as matrix inversion, solving linear systems, and eigenvalue problems. Recognizing how matrix components interact through arithmetic operations is critical to understanding broader linear algebra concepts.