Problem 16 - College Algebra

Question

For the following exercises, use the matrices below to perform scalar multiplication. $$ A=\left[\begin{array}{cc}{4} & {6} \\ {13} & {12}\end{array}\right], B=\left[\begin{array}{cc}{3} & {9} \\ {21} & {12} \\ {0} & {64}\end{array}\right], C=\left[\begin{array}{cccc}{16} & {3} & {7} & {18} \\\ {90} & {5} & {3} & {29}\end{array}\right], D=\left[\begin{array}{ccc}{18} & {12} & {13} \\ {8} & {14} & {6} \\ {7} & {4} & {21}\end{array}\right] $$ $$ \frac{1}{2} C $$

Step-by-Step Solution

Answer

The result of \( \frac{1}{2} C \) is \( \begin{bmatrix} 8 & 1.5 & 3.5 & 9 \\ 45 & 2.5 & 1.5 & 14.5 \end{bmatrix} \).

Step 1: Identify the Matrix and Scalar

We are asked to perform scalar multiplication on matrix C using the scalar \( \frac{1}{2} \). The matrix C is given as:\[C = \begin{bmatrix} 16 & 3 & 7 & 18 \ 90 & 5 & 3 & 29 \end{bmatrix}\] and the scalar is \( \frac{1}{2} \).

Step 2: Apply the Scalar to Each Element

To perform the scalar multiplication, multiply each element of matrix C by \( \frac{1}{2} \).

Step 3: Calculate the Scalar Multiplication

Let's calculate each element: - First row: - \( \frac{1}{2} \times 16 = 8 \) - \( \frac{1}{2} \times 3 = 1.5 \) - \( \frac{1}{2} \times 7 = 3.5 \) - \( \frac{1}{2} \times 18 = 9 \)- Second row: - \( \frac{1}{2} \times 90 = 45 \) - \( \frac{1}{2} \times 5 = 2.5 \) - \( \frac{1}{2} \times 3 = 1.5 \) - \( \frac{1}{2} \times 29 = 14.5 \)

Step 4: Write the Resulting Matrix

After performing the calculations, the resulting matrix is:\[\frac{1}{2} C = \begin{bmatrix} 8 & 1.5 & 3.5 & 9 \ 45 & 2.5 & 1.5 & 14.5 \end{bmatrix}\]

Key Concepts

Matrix Operations

Matrix operations involve various techniques where you manipulate matrices to achieve specific outcomes. One of the most basic yet essential operations is scalar multiplication, where each element of a matrix is multiplied by a given number, called the scalar. Unlike other matrix operations such as matrix addition or multiplication, scalar multiplication is straightforward because it doesn't require matrices to conform to specific size rules.

• In scalar multiplication, you simply multiply every element in the matrix by the scalar.
• Each element responds individually, so the resulting size of the matrix remains unchanged.
• It's an element-wise process and important for rescaling or adjusting the magnitude of numbers within a matrix.

For instance, multiplying matrix C by 0.5 (scalar), every entry in C is divided by 2, transforming the original matrix into a new one with each value halved. This kind of manipulation is crucial when dealing with matrices in linear algebra, simplifying complex computations or adjusting datasets for analysis.

Element-Wise Multiplication

Element-wise multiplication is a process where each element of a matrix is treated individually during operations. In the context of scalar multiplication, it particularly means multiplying each matrix element by the scalar, rather than any interaction between different rows or columns.
This is a key concept because it separates scalar multiplication from other types of multiplication like matrix-to-matrix multiplication where you usually deal with dot products or other complex interactions.

• To perform element-wise multiplication correctly, ensure every item in the matrix is calculated independently with the scalar.
• It's a simple, direct approach that ensures each scalar-multiplied value maintains the proportioning of the original matrix.

During our exercise, each number in matrix C was multiplied by 0.5, showcasing that every element individually reacts to the calculation, resulting in a simplified or adjusted matrix that can then be utilized in further mathematical analysis.

Algebraic Manipulation

Algebraic manipulation is the broader context in which matrix operations and element-wise multiplication occur. It involves rearranging and simplifying expressions to make them easier to work with or to solve given problems. Matrices, in a mathematical sense, simplify complex problems by organizing data in rows and columns. Performing algebraic manipulations on matrices often involves

• Breaking down operations into simpler steps, like we did with scalar multiplication.
• Simplifying matrix equations through basic algebraic rules, which help in solving equations more effectively.

Scalar multiplication is a perfect example of algebraic manipulation because it simplifies matrices for more advanced operations or analysis. For instance, dividing each element of a matrix C by two through scalar multiplication directly results in a new matrix, creating a different perspective on the same set of data—readying it for further algebraic processing or application. Through mastery of these manipulations, you're better equipped to handle more challenging algebraic problems in matrix studies.