Problem 14 - College Algebra

Question

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

Step-by-Step Solution

Answer

The resulting matrix after multiplying matrix B by -2 is \(\begin{bmatrix} -6 & -18 \\ -42 & -24 \\ 0 & -128 \end{bmatrix}\).

Step 1: Understand Scalar Multiplication

Scalar multiplication involves multiplying every entry in the matrix by a scalar (constant). For this exercise, the scalar is \(-2\) and the matrix to be multiplied is \(B\).

Step 2: Write Original Matrix B

Matrix \(B\) is given as: \[B = \begin{bmatrix} 3 & 9 \ 21 & 12 \ 0 & 64 \end{bmatrix}\]

Step 3: Multiply Each Element by Scalar -2

Multiply each element of matrix \(B\) by \(-2\):- First row, first column: \(3 \times (-2) = -6\)- First row, second column: \(9 \times (-2) = -18\)- Second row, first column: \(21 \times (-2) = -42\)- Second row, second column: \(12 \times (-2) = -24\)- Third row, first column: \(0 \times (-2) = 0\)- Third row, second column: \(64 \times (-2) = -128\)

Step 4: Write the Resulting Matrix

The matrix after scalar multiplication by \(-2\) is: \[-2B = \begin{bmatrix} -6 & -18 \ -42 & -24 \ 0 & -128 \end{bmatrix}\]

Key Concepts

Matrices and Their Structure

A matrix is a rectangular grid of numbers arranged in rows and columns. Each number in a matrix is called an element or entry. Matrices are a foundational aspect of linear algebra and are used to solve systems of linear equations, among many other applications in various fields such as physics, computer science, and economics.

Matrices can be any size, from 1x1 to any larger dimensions, such as 2x2, 3x3, or even larger. The size of a matrix is defined by the number of rows and columns it has. For example, a matrix with two rows and three columns is a 2x3 matrix. Knowing this structure helps in carrying out different operations, such as addition, subtraction, and multiplication, involving matrices.

• The horizontal arrays of elements are called rows.
• The vertical arrays of elements are called columns.
• An individual element is usually defined by its row and column position, using an index.

Understanding this basic structure is crucial when working with matrices and performing operations like scalar multiplication.

Scalar Multiplication of a Matrix

Scalar multiplication involves multiplying every element of a matrix by a single number, known as a scalar. This operation is fundamental and helps in modifying the scaling and dimensions within matrix operations. Let's delve into this basic yet powerful tool in linear algebra.

In scalar multiplication, you take each element of the matrix and multiply it by the scalar. This is simpler than it might sound! For example, if our scalar is \(-2\), each element of the matrix \(B\) would be multiplied by \(-2\). This means that every number within the matrix becomes \(-2\) times its original value.

• Start by identifying the matrix you wish to multiply.
• Multiply each entry individually by the scalar value.
• The resulting matrix will have the same dimensions as the original matrix.

Performing this operation allows for quick and easy manipulation of data within the matrix, leading to significant changes in how the data behaves within various computations. Remember, while the scalar changes the values of the elements, it does not change the dimensionality of the matrix itself.

Basic Matrix Operations

Matrix operations are fundamental tools in algebra that include addition, subtraction, and multiplication (both scalar and matrix multiplication). They allow for the transformation and manipulation of data represented in matrix form. These operations are essential for both theoretical mathematical developments and practical applications, such as in computer graphics, engineering, and more.

Here, we focus on the contexts required to understand scalar multiplication, yet similar insights will apply when dealing with other matrix operations. Matrix operations require matrices to have certain dimensions so that they can be combined effectively:

• Addition/Subtraction: Operate element-wise, meaning each element in one matrix is added or subtracted from the respective element in another matrix. For this to happen, the matrices must have the same dimensions.
• Scalar Multiplication: As covered, involves changing the scale of each element by multiplying them all by a constant number.
• Matrix Multiplication: Requires more complex rules, such as the number of columns in the first matrix needing to equal the number of rows in the second matrix. This operation results in a new matrix with different dimensions.

Mastering these operations paves the way for more complex problem-solving strategies, enabling you to use matrices effectively in data transformations, optimizations, and even in solving equations using matrices.