Problem 7 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{bmatrix} -1 & -5 \\ 2 & 3 \end{bmatrix} \).

Step 1: Subtract Matrix B from Matrix A

First, perform the subtraction of matrix B from matrix A. The matrices are: \[ A = \begin{bmatrix} 1 & -2 \ 3 & 4 \end{bmatrix}, \, B = \begin{bmatrix} 2 & -3 \ 5 & -1 \end{bmatrix} \]The subtraction \( A - B \) is computed by subtracting corresponding elements: \[ A - B = \begin{bmatrix} 1 - 2 & -2 - (-3) \ 3 - 5 & 4 - (-1) \end{bmatrix} = \begin{bmatrix} -1 & 1 \ -2 & 5 \end{bmatrix} \].

Step 2: Subtract Matrix C from the Result

Now, subtract matrix C from the earlier result. Matrix C is: \[ C = \begin{bmatrix} 0 & 6 \ -4 & 2 \end{bmatrix} \]Take the result from Step 1 and subtract C: \[ (A - B) - C = \begin{bmatrix} -1 & 1 \ -2 & 5 \end{bmatrix} - \begin{bmatrix} 0 & 6 \ -4 & 2 \end{bmatrix} \]Calculate each element:\[ (A-B)-C = \begin{bmatrix} -1 - 0 & 1 - 6 \ -2 - (-4) & 5 - 2 \end{bmatrix} = \begin{bmatrix} -1 & -5 \ 2 & 3 \end{bmatrix} \].

Key Concepts

Matrix Operations: Understanding Subtraction

Matrix operations are fundamental tools in mathematics and can be used in a wide variety of applications. Among these operations, matrix subtraction is an essential skill to master. In matrix subtraction, just like in matrix addition, you perform the operation element-wise. This means that each corresponding element of the matrices is subtracted from one another.

To get a bit more detailed, let's assume you have two matrices, say Matrix A and Matrix B, both of the same dimensions. The subtraction of matrices, denoted as A - B, involves subtracting each element in Matrix B from the corresponding element in Matrix A. Keep in mind, the matrices must have the same size; otherwise, the operation is undefined.

• The subtraction is done by calculating a new matrix where each element is found by taking the element in the same position from Matrix B and subtracting it from the corresponding element in Matrix A.
• This operation maintains the same dimensions as the original matrices.

For example, \[A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}, \, B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \]would result in,\[A - B = \begin{bmatrix} a_{11} - b_{11} & a_{12} - b_{12} \ a_{21} - b_{21} & a_{22} - b_{22} \end{bmatrix}\]This operation is relatively straightforward but serves as a foundational concept in more complex manipulations in matrix algebra.

Elementary Matrix Algebra: Introducing the Basics

Elementary matrix algebra involves understanding simple operations and definitions that apply to matrices. Besides basic arithmetic on individual elements, such as addition and subtraction, you're dealing with square structures with clearly defined rules. Algebra concerning matrices can be used to solve linear equations, perform transformations, and represent systems effectively.

• Matrix Elements: Each item in a matrix is known as an element. When performing operations, each element is treated similarly to numbers in standard arithmetic.
• Dimensions: Matrices are defined using their dimensions, often represented as 'm x n', where 'm' is the number of rows and 'n' is the number of columns.

One should remember that the operations such as addition and subtraction only apply to matrices with the same dimensions. This ensures that each element in the first matrix has a corresponding element in the second matrix, which is essential for operations like subtraction.

Remember that matrix algebra is the backbone of computational mathematics and is extensively used in computer graphics, data analysis, and virtually any field requiring complex calculation and data processing.

Algebraic Computation: Executing the Math

Algebraic computation in the context of matrices involves carrying out mathematical operations with precision and accuracy. It depends heavily on understanding both the possibilities and the limitations of matrix operations and handling them effectively.

When subtracting matrices, Algebraic computation requires careful attention to detail to ensure every element is treated correctly. For example, in the given exercise, the computation involved:

• Two matrices A and B, where the subtraction executed as \( A - B \) before moving to perform the operation \((A-B) - C\).
• Using the resulting matrix from the first operation as the starting point for the next subtraction involving Matrix C.

When proceeding with these operations, it's crucial to proceed through each calculation methodically, paying close attention to each subtraction you perform on corresponding elements continuity.

These step-by-step approaches not only ensure the correct results but also build a robust understanding of matrix computations, facilitating learning and application in more complex mathematical challenges. By honing these basic skills, you're set to tackle more advanced problems, as these are building blocks for techniques like matrix multiplication and solving systems of linear equations using matrix methods.