Problem 2 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{pmatrix} 2 & -9 \\ 9 & -3 \end{pmatrix} \).

Step 1: Identify the Matrices

Before computing the difference between two matrices, identify matrices B and C from the provided data: Matrix B is \( \begin{pmatrix} 2 & -3 \ 5 & -1 \end{pmatrix} \) and Matrix C is \( \begin{pmatrix} 0 & 6 \ -4 & 2 \end{pmatrix} \).

Step 2: Set Up the Subtraction

In matrix subtraction, subtract corresponding elements of one matrix from the other. Arrange matrices B and C for subtraction: \[B - C = \begin{pmatrix} 2 & -3 \ 5 & -1 \end{pmatrix} - \begin{pmatrix} 0 & 6 \ -4 & 2 \end{pmatrix}.\]

Step 3: Subtract the Corresponding Elements

Subtract each element of matrix C from the corresponding element in matrix B:\[B - C = \begin{pmatrix} 2-0 & -3-6 \ 5-(-4) & -1-2 \end{pmatrix} = \begin{pmatrix} 2 & -9 \ 9 & -3 \end{pmatrix}.\]

Step 4: Verify the Calculation

Ensure that each subtraction was performed correctly: - Top-left: \(2 - 0 = 2\)- Top-right: \(-3 - 6 = -9\)- Bottom-left: \(5 - (-4) = 9\)- Bottom-right: \(-1 - 2 = -3\)The resulting matrix is correctly calculated as \( \begin{pmatrix} 2 & -9 \ 9 & -3 \end{pmatrix} \).

Key Concepts

Matrix Operations

Matrix operations are essential tools in mathematics, used broadly in areas such as algebra, computer graphics, and data science. One of the basic operations is matrix subtraction. Just like with numbers, matrices can be added or subtracted from one another as long as they have the same dimensions. This means each matrix must have the same number of rows and columns.

Matrix subtraction involves taking two matrices of identical size and subtracting each element of one matrix from the corresponding element in the other matrix. For example, with matrices B and C, both are 2x2 matrices. We can subtract them by following these steps:

• Identify the corresponding elements in matrix B and matrix C.
• Subtract each pair, element by element.

This approach to matrix subtraction provides a useful method to solve problems in applied mathematics and physics, where differences in data represented in matrix form are common.

By mastering these operations, students can further their understanding of more complex mathematical concepts like matrix multiplication or finding the inverse of a matrix.

Algebraic Techniques

Algebraic techniques in matrix operations are crucial for solving equations systematically. In our example, subtracting two matrices is a straightforward task that improves algebraic fluency. This process strengthens foundational skills necessary for more advanced practices involving matrices.

To subtract matrices, one must understand the concept of corresponding elements, which mirrors the arithmetic operation with variables. The steps involved harness simple algebraic principles:

• Take two matrices of equal size.
• Perform element-wise subtraction where each element in one matrix subtracts the matching element in the other.

For instance, applying this to matrices B and C, we find:\[ B - C = \begin{pmatrix} 2 & -3 \ 5 & -1 \end{pmatrix} - \begin{pmatrix} 0 & 6 \ -4 & 2 \end{pmatrix} = \begin{pmatrix} 2 & -9 \ 9 & -3 \end{pmatrix} \] This result is achieved by pure arithmetic principles applied in a step-by-step manner, instilling confidence and mathematical reasoning.

Such algebraic techniques make mathematics less intimidating and help students see the logical steps involved in both solving equations and performing operations.

Educational Mathematics

Educational mathematics, as taught in schools and colleges, helps students understand complex concepts with simpler exercises first. Matrix subtraction is a perfect example of this educational approach. By starting with basic operations, students learn the properties and rules that govern matrices.

Exercises that focus on subtraction teach students to:

• Identify matrix dimensions.
• Perform basic arithmetic operations.
• Recognize patterns and rows within data sets.

These concepts ground their knowledge before moving to intricate applications, like transformations and solving systems of equations.

Incorporating matrix operations in an educational setting provides hands-on experience with abstract numbers and offers insight into structure and order within algebraic systems. When tackled incrementally, topics like matrix subtraction help demystify what can often be a challenging subject, empowering students to learn progressively and eventually master more advanced topics.