Problem 3 - Algebra for College Students

Question

For Problems \(1-12\), compute the indicated matrix by using the following matrices: \(\begin{array}{ll} A=\left[\begin{array}{rr} 1 & -2 \\ 3 & 4 \end{array}\right] & B=\left[\begin{array}{lr} 2 & -3 \\ 5 & -1 \end{array}\right] \\ C=\left[\begin{array}{rr} 0 & 6 \\ -4 & 2 \end{array}\right] & D=\left[\begin{array}{rr} -2 & 3 \\ 5 & -4 \end{array}\right] \\ E=\left[\begin{array}{lr} 2 & 5 \\ 7 & 3 \end{array}\right] & \end{array}\) $$ 3 C+D $$

Step-by-Step Solution

Answer

The matrix is \( \begin{bmatrix} -2 & 21 \\ -7 & 2 \end{bmatrix} \).

Step 1: Scalar Multiplication of Matrix C

First, we need to multiply each element of matrix \( C \) by the scalar 3. Matrix \( C \) is \( \begin{bmatrix} 0 & 6 \ -4 & 2 \end{bmatrix} \). Multiply each element by 3: \(\begin{bmatrix} 0 \times 3 & 6 \times 3 \ -4 \times 3 & 2 \times 3 \end{bmatrix} = \begin{bmatrix} 0 & 18 \ -12 & 6 \end{bmatrix} \).

Step 2: Add Matrix D

Next, we add matrix \( D \) to the result from Step 1. Matrix \( D \) is \( \begin{bmatrix} -2 & 3 \ 5 & -4 \end{bmatrix} \). So, we add corresponding elements: \( \begin{bmatrix} 0 & 18 \ -12 & 6 \end{bmatrix} + \begin{bmatrix} -2 & 3 \ 5 & -4 \end{bmatrix} = \begin{bmatrix} 0 + (-2) & 18 + 3 \ -12 + 5 & 6 + (-4) \end{bmatrix} = \begin{bmatrix} -2 & 21 \ -7 & 2 \end{bmatrix} \).

Key Concepts

Scalar Multiplication

Scalar multiplication is an essential operation in matrix algebra where each element of a matrix is multiplied by a single number, known as a scalar. This operation is straightforward yet powerful in the way it scales a matrix. For example, given the matrix \( C = \begin{bmatrix} 0 & 6 \ -4 & 2 \end{bmatrix} \), and a scalar, such as 3, we proceed by multiplying each element of the matrix by 3:

• \(0 \times 3 = 0\)
• \(6 \times 3 = 18\)
• \(-4 \times 3 = -12\)
• \(2 \times 3 = 6\)

So, after scalar multiplication, we obtain the matrix \( \begin{bmatrix} 0 & 18 \ -12 & 6 \end{bmatrix} \). Remember, scalar multiplication does not alter the shape of the matrix; it solely affects the magnitude of its elements.

Matrix Addition

Matrix addition involves summing matrices of the same dimensions by adding their corresponding elements. For example, consider matrices \( \begin{bmatrix} a_{1} & b_{1} \ c_{1} & d_{1} \end{bmatrix} \) and \( \begin{bmatrix} a_{2} & b_{2} \ c_{2} & d_{2} \end{bmatrix} \). The sum is given by:

• \((a_{1} + a_{2})\)
• \((b_{1} + b_{2})\)
• \((c_{1} + c_{2})\)
• \((d_{1} + d_{2})\)

In our problem, after finding \( 3C \), we have \( \begin{bmatrix} 0 & 18 \ -12 & 6 \end{bmatrix} \). We then add this result to matrix \( D = \begin{bmatrix} -2 & 3 \ 5 & -4 \end{bmatrix} \):

• \(0 + (-2) = -2\)
• \(18 + 3 = 21\)
• \(-12 + 5 = -7\)
• \(6 + (-4) = 2\)

Thus resulting in the matrix \( \begin{bmatrix} -2 & 21 \ -7 & 2 \end{bmatrix} \). Ensure both matrices are of the same dimensions before attempting addition.

Elementary Algebra

Elementary algebra serves as the foundation for understanding the basic operations and properties that apply to entire classes of numbers and variables. These operations underpin skills necessary for more advanced algebraic concepts.
When considering matrix operations like those in this exercise, remember:

• Just like algebraic variables, matrices can be manipulated with operations such as addition, subtraction, and multiplication.
• Scalars are akin to coefficients in algebra. They multiply every term within the matrix, similar to how a coefficient distributes across variables.
• Matrix dimensions must match for addition and subtraction, analogous to the requirement for combining like terms in algebra.

Understanding these elementary properties allows one to build confidence in handling algebraic expressions within matrices effectively, and vice versa.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It forms the backbone of various fields like computer science, engineering, physics, and statistics.
Within linear algebra, we saw two key operations: scalar multiplication and matrix addition. These operations illustrate concepts like vector space scaling and translating, ultimately providing deeper insights into the structure and behavior of linear systems.

• Scalar multiplication stretches or shrinks vectors, similar to changes in amplitude and magnitude.
• Matrix addition models the combination of multiple linear transformations or system states.

The exercise helps highlight linear algebra's real-world applications, showcasing how such operations facilitate problem-solving in complex systems. For students, mastering these tools in linear algebra will enhance understanding in both theoretical and practical domains.