Problem 17 - College Algebra

Question

For the following exercises, use the matrices below to perform scalar multiplication. $$ A=\left[\begin{array}{cc}{4} & {6} \\ {13} & {12}\end{array}\right], B=\left[\begin{array}{cc}{3} & {9} \\ {21} & {12} \\ {0} & {64}\end{array}\right], C=\left[\begin{array}{cccc}{16} & {3} & {7} & {18} \\\ {90} & {5} & {3} & {29}\end{array}\right], D=\left[\begin{array}{ccc}{18} & {12} & {13} \\ {8} & {14} & {6} \\ {7} & {4} & {21}\end{array}\right] $$ $$ 100 D $$

Step-by-Step Solution

Answer

The resulting matrix from 100D is \( \begin{bmatrix} 1800 & 1200 & 1300 \\ 800 & 1400 & 600 \\ 700 & 400 & 2100 \end{bmatrix} \).

Step 1: Understanding Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a constant value, known as the scalar. In this exercise, the matrix is \( D \) and the scalar is 100.

Step 2: Matrix D

The matrix \( D \) is given by:\[D = \begin{bmatrix} 18 & 12 & 13 \ 8 & 14 & 6 \ 7 & 4 & 21 \end{bmatrix}\]

Step 3: Perform Scalar Multiplication

Multiply each element of matrix \( D \) by 100:\[100D = \begin{bmatrix} 100 \times 18 & 100 \times 12 & 100 \times 13 \ 100 \times 8 & 100 \times 14 & 100 \times 6 \ 100 \times 7 & 100 \times 4 & 100 \times 21 \end{bmatrix}\]

Step 4: Calculate Products

Calculate each multiplication:- \(100 \times 18 = 1800\)- \(100 \times 12 = 1200\)- \(100 \times 13 = 1300\)- \(100 \times 8 = 800\)- \(100 \times 14 = 1400\)- \(100 \times 6 = 600\)- \(100 \times 7 = 700\)- \(100 \times 4 = 400\)- \(100 \times 21 = 2100\)

Step 5: Construct the Resulting Matrix

Assemble the results into the final matrix:\[100D = \begin{bmatrix} 1800 & 1200 & 1300 \ 800 & 1400 & 600 \ 700 & 400 & 2100 \end{bmatrix}\]

Key Concepts

Matrices

Matrices are organized collections of numbers laid out in rows and columns, creating a rectangular array. Each number in a matrix is called an element.
They are used extensively in various fields to solve mathematical problems, represent data, and model real-world systems.

• Matrices can have different sizes, determined by the number of rows and columns, also known as dimensions.
• A matrix with rows and columns is classified as an \(m \times n\) matrix, where \(m\) is the number of rows, and \(n\) is the number of columns.
• Elements within a matrix are typically represented by their position using two indices, one for the row and one for the column. For example, an element \(a_{ij}\) represents the entry located at the \(i\)-th row and \(j\)-th column.

Learning about matrices is crucial because they provide a simple way to work with linear equations, transformations, and multi-dimensional data.
A clear understanding of matrices enables solving complex mathematical problems much more efficiently.

Matrix Operations

Matrix operations are powerful tools in mathematics that allow us to manipulate and compute with matrices. Scalar multiplication, the operation in focus here, is one of the basic matrix operations.

• Scalar Multiplication: Each element of the matrix is multiplied by a constant number, known as the scalar. This is what happens in the exercise where matrix \(D\) is multiplied by 100.
• Addition and Subtraction: Matrices of the same size can be added or subtracted by simply adding or subtracting their corresponding elements. This requires the matrices to have the exact same dimensions.
• Matrix Multiplication: In contrast to scalar multiplication, matrix multiplication involves multiplying rows from the first matrix by columns from the second. This operation results in a new matrix.

Each operation must follow specific rules and conditions for successful execution. Understanding these operations lays the groundwork for more advanced applications, including solving system of equations and transformations in space.

Algebra

Algebra often forms the backbone for understanding matrix operations, providing the necessary framework and rules for manipulating these numbers and symbols. In this context, algebra provides the skills needed to handle matrices effectively.

• Constants and Variables: In matrices, numbers can be constants (like the scalar 100) that remain unchanged or variables that can change values.
• Equations and Expressions: Algebra helps formulate equations that matrices can easily solve. These involve expressions that can contain both numbers and variables.
• Distributive Property: Scalar multiplication follows the distributive property, ensuring that when you multiply a matrix by a scalar, each element is multiplied separately, as shown in the exercise solution.

Algebraic skills enhance your ability to solve complicated problems by breaking them into simpler, manageable parts. It also makes matrix manipulation intuitive, as you rely on algebraic properties to understand the interactions within and between matrices.