Problem 17 - College Algebra

Question

Use the matrices below to perform scalar multiplication. \(A=\left[\begin{array}{rr}4 & 6 \\ 13 & 12\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 21 & 12 \\ 0 & 64\end{array}\right], C=\left[\begin{array}{rrrr}16 & 3 & 7 & 18 \\ 90 & 5 & 3 & 29\end{array}\right], D=\left[\begin{array}{rrr}18 & 12 & 13 \\ 8 & 14 & 6 \\\ 7 & 4 & 21\end{array}\right]\) \(100 D\)

Step-by-Step Solution

Answer

Multiply each element of matrix D by 100 to get \( 100D = \left[\begin{array}{rrr}1800 & 1200 & 1300 \\ 800 & 1400 & 600 \\\ 700 & 400 & 2100\end{array}\right] \).

Step 1: Understanding Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a scalar (a constant value). For the matrix \( D \) and scalar 100, each element of \( D \) is multiplied by 100.

Step 2: Set Up Matrix D

Matrix \( D \) is given as \( \left[\begin{array}{rrr}18 & 12 & 13 \ 8 & 14 & 6 \ 7 & 4 & 21\end{array}\right] \). We'll multiply each of these elements by 100.

Step 3: Multiply First Row by 100

Multiply each element of the first row of matrix \( D \), \( \left[18, 12, 13\right] \), by 100: \( 18 \times 100 = 1800 \), \( 12 \times 100 = 1200 \), \( 13 \times 100 = 1300 \). The resulting first row is \( \left[1800, 1200, 1300\right] \).

Step 4: Multiply Second Row by 100

Multiply each element of the second row of matrix \( D \), \( \left[8, 14, 6\right] \), by 100: \( 8 \times 100 = 800 \), \( 14 \times 100 = 1400 \), \( 6 \times 100 = 600 \). The resulting second row is \( \left[800, 1400, 600\right] \).

Step 5: Multiply Third Row by 100

Multiply each element of the third row of matrix \( D \), \( \left[7, 4, 21\right] \), by 100: \( 7 \times 100 = 700 \), \( 4 \times 100 = 400 \), \( 21 \times 100 = 2100 \). The resulting third row is \( \left[700, 400, 2100\right] \).

Step 6: Combine the Results

Combine all the results to form the new matrix after scalar multiplication: \( 100D = \left[\begin{array}{rrr}1800 & 1200 & 1300 \ 800 & 1400 & 600 \ 700 & 400 & 2100\end{array}\right] \).

Key Concepts

matrix multiplication

Matrix multiplication is a fundamental operation in mathematics, especially in the context of linear algebra. The operation itself significantly differs from multiplying individual numbers. When multiplying two matrices, you take rows from the first matrix and columns from the second, computing the sum of element-wise products. This operation results in a new matrix.

Some key points include:

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• The resulting matrix will have the dimensions of the rows of the first matrix and the columns of the second matrix.
• Matrix multiplication is not commutative, meaning that the order in which you multiply matrices matters.

Understanding matrix multiplication aids in grasping more complex matrix operations.

matrix operations

Matrix operations encompass a range of manipulations, including addition, subtraction, and multiplication. Scalar multiplication, where each element of a matrix is multiplied by a scalar, is one of the simplest forms of matrix operations.

The following operations are essential for manipulating matrices:

• Addition: Only applicable for matrices of the same dimension, achieved by adding corresponding elements.
• Subtraction: Similar to addition, but involves subtracting corresponding elements of matrices of the same size.
• Scalar Multiplication: Every element in the matrix is multiplied by a scalar (constant value), as demonstrated in the exercise.
• Matrix Multiplication: More complex and requires alignment of matrix dimensions, leading to a new matrix based on row and column interactions.

Matrix operations form the foundation of many mathematical and engineering applications.

matrices

Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. They serve as vital tools in mathematics to perform a variety of operations.

Key characteristics of matrices include:

• Dimension: Defined by the number of rows and columns (m x n).
• Elements: Individual items within a matrix, located based on their row and column position.
• Square Matrix: A matrix where the number of rows equals the number of columns.
• Identity Matrix: A special kind of square matrix with 1s on the diagonal and 0s elsewhere.

Matrices are instrumental in solving linear equations, transformations, and in representing data in various fields like physics and computer science.

algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the context of matrices, algebra involves understanding how these symbolic arrays interact and transform through various operations.

Here's how algebra intertwines with matrices:

• Matrices use algebraic structures to solve equations and perform transformations in a simplified and structured form.
• Matrix algebra assists in deriving solutions to simultaneous equations, which are common in several scientific applications.
• Algebraic expressions apply to matrices, following specific rules that differ slightly from arithmetic operations on numbers.

Mastering algebraic concepts in matrices opens the door to tackling real-world problems involving systems of equations and data transformations.