Problem 16 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{bmatrix} 8 & 1.5 & 3.5 & 9 \\ 45 & 2.5 & 1.5 & 14.5 \end{bmatrix} \).

Step 1: Identify the scalar and matrix

In this exercise, we are given the matrix \( C \) and a scalar \( \frac{1}{2} \). The goal is to multiply every element in matrix \( C \) by the scalar \( \frac{1}{2} \).

Step 2: Write down matrix C

Matrix \( C \) is: \[C = \begin{bmatrix} 16 & 3 & 7 & 18 \ 90 & 5 & 3 & 29 \end{bmatrix}\] We will use this matrix for our scalar multiplication.

Step 3: Scalar multiplication of row 1

Take each element of the first row of matrix \( C \) and multiply by the scalar \( \frac{1}{2} \): \(\frac{1}{2} \times 16 = 8 \) \(\frac{1}{2} \times 3 = 1.5 \) \(\frac{1}{2} \times 7 = 3.5 \) \(\frac{1}{2} \times 18 = 9 \)

Step 4: Scalar multiplication of row 2

Take each element of the second row of matrix \( C \) and multiply by the scalar \( \frac{1}{2} \): \( \frac{1}{2} \times 90 = 45 \) \( \frac{1}{2} \times 5 = 2.5 \) \(\frac{1}{2} \times 3 = 1.5 \) \(\frac{1}{2} \times 29 = 14.5 \)

Step 5: Assemble the resulting matrix

After performing the scalar multiplications for each element, we obtain the new matrix: \[\frac{1}{2} C = \begin{bmatrix} 8 & 1.5 & 3.5 & 9 \ 45 & 2.5 & 1.5 & 14.5 \end{bmatrix}\]

Key Concepts

matrix operations

Matrix operations are foundational in linear algebra and essential for manipulating data in various fields such as physics, engineering, and computer science. These operations can include addition, subtraction, scalar multiplication, and even finding the determinant of a matrix. Each operation serves a different purpose and has its own set of rules:

• Addition and Subtraction: These operations require matrices of the same dimensions, where corresponding elements are added or subtracted from each other.
• Scalar Multiplication: Involves multiplying each element of a matrix by a scalar (a single number).
• Matrix Multiplication: More complex, this operation involves multiplying rows by columns, and the resulting matrix has different dimensions.

Understanding these operations is crucial in solving linear systems and performing transformations in vector spaces. Practical applications are vast, such as solving complex equations or modeling data structures.

reduced matrix

A reduced matrix typically refers to a matrix that has been simplified in some way. A common form of reduction is row reduction, where a matrix is transformed into row-echelon form or reduced row-echelon form, often called Gaussian elimination. This reduction is significant when solving systems of linear equations:

• Row-Echelon Form: Simply put, this form displays zeros below leading coefficients, which appear as ones on the diagonal.
• Reduced Row-Echelon Form: Here, all entries in a leading position are ones, and other entries in the column containing a leading one are zero.

Reduction makes matrices easier to work with, transforming complex equations into solvable formats. By breaking down matrices into simpler forms, it is easier to discern patterns or solutions to equations they represent.

matrix scalar multiplication

Matrix scalar multiplication involves multiplying every element of a matrix by a scalar, which is a constant number. For example, if you have a matrix and a scalar value of \( \frac{1}{2} \), you multiply each number in the matrix by \( \frac{1}{2} \). This operation scales the matrix, effectively enlarging or shrinking its values. Here's how to perform scalar multiplication:

• Identify your matrix and scalar.
• Multiply every element within the matrix by the scalar.
• Ensure the resulting matrix maintains the same dimensions.

As seen in the Original Step by Step Solution, Matrix \( C \) is multiplied by \( \frac{1}{2} \), halving each of its elements. The process demonstrates how scalar multiplication alters the value of each element, yet retains the structure of the matrix.

matrices

Matrices are rectangular arrays of numbers organized into rows and columns. They are pivotal in various scientific and engineering fields due to their ability to efficiently handle large datasets and complex systems of equations. A few key features about matrices include:

• Dimensions: Described by rows and columns, like "2x2" or "3x2", indicating the number of rows followed by the number of columns.
• Elements: The specific numbers within a matrix, each identified by its position, such as element \( a_{11} \) being the first row, first column entry.
• Types: They can be square, symmetric, diagonal, identity, or zero matrices, each serving different purposes and properties.

Understanding the structure and characteristics of matrices is fundamental in mathematics and their applications make solving practical problems more efficient and streamlined. Whether through transformations or data manipulations, matrices effectively simplify complex multi-variable systems.