Problem 12 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The resulting matrix is \(5A = \begin{bmatrix} 20 & 30 \\ 65 & 60 \end{bmatrix}.\)

Step 1: Understanding Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a given scalar. In this exercise, we need to multiply matrix \(A\) by the scalar \(5\).

Step 2: Identifying Matrix A

Matrix \(A\) is given as \[A = \begin{bmatrix} 4 & 6 \ 13 & 12 \end{bmatrix}.\]This is a \(2 \times 2\) matrix.

Step 3: Multiply Each Element by 5

To perform the scalar multiplication, multiply each element in matrix \(A\) by \(5\):- The top left element: \(5 \times 4 = 20\)- The top right element: \(5 \times 6 = 30\)- The bottom left element: \(5 \times 13 = 65\)- The bottom right element: \(5 \times 12 = 60\)

Step 4: Construct the Resulting Matrix

After performing the multiplication for each element, construct the new matrix from the results: \[5A = \begin{bmatrix} 20 & 30 \ 65 & 60 \end{bmatrix}.\]

Key Concepts

Matrix Operations

Matrix operations are fundamental actions that can be performed on matrices, including addition, subtraction, and multiplication. In this particular exercise, we focus on scalar multiplication, a straightforward yet powerful operation.

When you perform scalar multiplication, you multiply every element of a matrix by a specific number, called the scalar. For example, given a matrix \( A \) and a scalar \( k \), you create a new matrix by multiplying each element of \( A \) by \( k \). This operation is essential for modifying the scale of the matrix's data without changing their relative structure.

It's important to remember that, unlike matrix multiplication, scalar multiplication maintains the original order of the rows and columns. The scalar simply scales all the entries evenly, which is particularly useful when you need to amplify or attenuate the entire dataset represented by the matrix. This operation serves as a building block for more complex tasks in linear algebra.

Matrix Arithmetic

Matrix arithmetic involves various types of operations such as addition, subtraction, and multiplication, including scalar multiplication, by which we alter and manipulate matrices.

In matrix arithmetic, scalar multiplication ensures uniform scaling across all matrix elements. Here are some key points:

• **Commutativity:** Unlike normal arithmetic, matrix multiplication is not commutative. However, scalar multiplication is commutative: \( k \times A = A \times k \).
• **Distributivity:** Scalar multiplication is distributive with respect to matrix addition; thus, \( k \times (A+B) = k \times A + k \times B \).
• **Associativity:** Scalar multiplication is associative with another scalar: \( (m\times k) \times A = m \times (k \times A) \).

Understanding these properties helps you predict and compute the results of more complicated operations effectively.

These operations are crucial for processes like linear transformations, where matrices are used to map vectors from one space to another. Efficiently mastering scalar multiplication will pave the way for confidence in tackling complex matrix arithmetic.

Elementary Linear Algebra

Elementary linear algebra deals with the study of vectors, matrices, and their transformations. Scalar multiplication is one of the foundational operations in this area that enables you to explore how linear systems behave under transformations, which are crucial in various applications such as physics, engineering, and computer science.

In the context of linear algebra, matrices can represent systems of linear equations. When performing scalar multiplication:

• The result doesn't alter the solution space of a linear system but scales the values of potential solutions.
• Scalar multiplication helps in simplifying matrix equations, making them easier to solve or interpret.
• This operation is instrumental in calculating determinants and eigenvectors, key concepts in more advanced linear algebra topics.

By mastering elementary scalar multiplication, you form the foundation needed to delve deeper into more advanced linear algebra concepts and their applications, ensuring a solid understanding of matrix behavior in various mathematical contexts.