Problem 8 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The result is \( \begin{bmatrix} 6 & -1 \\ 7 & 6 \end{bmatrix} \).

Step 1: Compute D-E

First, subtract matrix E from matrix D. To do this, subtract each element of E from the corresponding element of D: \(D = \begin{bmatrix} -2 & 3 \5 & -4 \end{bmatrix}, E = \begin{bmatrix} 2 & 5 \7 & 3 \end{bmatrix} \)Subtract E from D element-wise: \[-2 - 2 = -4, \quad 3 - 5 = -2\]\[5 - 7 = -2, \quad -4 - 3 = -7\]Thus, \(D-E = \begin{bmatrix} -4 & -2 \-2 & -7 \end{bmatrix}\).

Step 2: Compute B-(D-E)

Next, subtract the result from Step 1 from matrix B. Matrix B is given as:\(B = \begin{bmatrix} 2 & -3 \5 & -1 \end{bmatrix} \)Subtract \(D-E\) from B element-wise: \[2 - (-4) = 2 + 4 = 6, \quad -3 - (-2) = -3 + 2 = -1\]\[5 - (-2) = 5 + 2 = 7, \quad -1 - (-7) = -1 + 7 = 6\]Thus, \(B - (D-E) = \begin{bmatrix} 6 & -1 \7 & 6 \end{bmatrix}\).

Key Concepts

Matrix Operations

Matrix operations are fundamental in the field of mathematics, especially in linear algebra. They include a variety of processes such as addition, subtraction, multiplication, and sometimes finding the inverse of matrices. Each operation follows specific rules that can differ from operations on scalar numbers.
For subtraction, each entry in the resulting matrix is derived from element-wise subtraction of corresponding entries from the involved matrices.

• Matrix addition and subtraction require matrices to be of the same size.
• Every entry in the first matrix has a corresponding entry in the second matrix, and operations are performed pair-wise.
• The resulting matrix maintains the same dimensions as the original matrices.

Understanding matrix operations lays the groundwork for tackling more complex problems in linear algebra, such as solving systems of equations or performing transformations in geometry.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations, as well as systems of linear equations. It provides a framework for understanding the structure and solutions of linear systems, enabling vast applications in science and engineering. At its core, linear algebra deals with

• vectors and their operations,
• matrices and matrix operations,
• linear transformations, and
• solutions to systems of linear equations.

Within the context of matrix operations, linear algebra facilitates ways to perform various operations including solving for variables and computing matrix determinants and inverses. Matrix subtraction, as a part of linear algebra, is used to manipulate and solve equations or compare datasets fit into matrix forms.

Element-wise Subtraction

Element-wise subtraction is a specific type of matrix subtraction where each element in the matrix is subtracted from its corresponding element in another matrix. This operation is straightforward and requires the matrices to be of matching dimensions.
Here's the basic procedure:

• Identify the corresponding elements in each matrix that need to be subtracted.
• Perform the subtraction operation on each pair of corresponding elements.
• Create a new matrix with the results of these subtractions.

For example, given two matrices:\[A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\]The element-wise subtraction \( A-B \) 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 element-wise approach ensures simplicity and clarity, as it mirrors the general arithmetic operations we perform on individual numbers.