Problem 13 - Contemporary Precalculus

Question

Evaluate the expression. $$\left(\begin{array}{l}5 \\\3\end{array}\right)+\left(\begin{array}{l}5 \\\2\end{array}\right)-\left(\begin{array}{l}6 \\\3\end{array}\right)$$

Step-by-Step Solution

Answer

Answer: The result of the expression is \(\left(\begin{array}{l}4 \\\2\end{array}\right)\).

Step 1: Identify the matrices

Identify and label the three matrices given in the expression. Matrix A: \(\left(\begin{array}{l}5 \\\3\end{array}\right)\) Matrix B: \(\left(\begin{array}{l}5 \\\2\end{array}\right)\) Matrix C: \(\left(\begin{array}{l}6 \\\3\end{array}\right)\)

Step 2: Add matrices A and B element-wise

Add corresponding elements of Matrix A and Matrix B. \((A_{11} + B_{11}, A_{21} + B_{21}) = (5+5, 3+2) = (10, 5)\) The result is a new matrix D: Matrix D: \(\left(\begin{array}{l}10 \\\5\end{array}\right)\)

Step 3: Subtract matrix C from matrix D element-wise

Subtract corresponding elements of Matrix D and Matrix C. \((D_{11} - C_{11}, D_{21} - C_{21}) = (10-6, 5-3) = (4, 2)\) The result is the final matrix E: Matrix E: \(\left(\begin{array}{l}4 \\\2\end{array}\right)\)

Step 4: Write the final answer

The result of the given expression is: $$\left(\begin{array}{l}5 \\\3\end{array}\right)+\left(\begin{array}{l}5 \\\2\end{array}\right)-\left(\begin{array}{l}6 \\\3\end{array}\right) = \left(\begin{array}{l}4 \\\2\end{array}\right)$$

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in matrix arithmetic that involves combining two matrices by adding their corresponding elements together. For two matrices to be added, they must have the same dimensions, meaning they must each have the same number of rows and columns. Let's break down the process for better understanding:

• Each element from Matrix A is added to the corresponding element in Matrix B.
• The resulting sum forms a new matrix, known as the resultant matrix.

For example, if we consider Matrix A and Matrix B from the original problem, we add as follows: \[\begin{pmatrix}A_{11} \A_{21}\end{pmatrix}+\begin{pmatrix}B_{11} \B_{21}\end{pmatrix}=\begin{pmatrix}A_{11} + B_{11} \A_{21} + B_{21}\end{pmatrix}\]This straightforward process allows us to compute element-wise sums, resulting in a new matrix that retains the dimensions of the original matrices.

Matrix Subtraction

Matrix subtraction, like matrix addition, involves element-wise operations and requires that the matrices involved have identical dimensions. Subtracting one matrix from another means taking each element of the first matrix and subtracting the corresponding element of the second matrix.

• Subtract element by element from the first matrix, the corresponding elements from the second matrix.
• The result will be a new matrix in which each element represents this difference.

For example, if we take Matrix D and subtract Matrix C from it as shown in the problem, the operation looks like this:\[\begin{pmatrix}D_{11} \D_{21}\end{pmatrix}-\begin{pmatrix}C_{11} \C_{21}\end{pmatrix}=\begin{pmatrix}D_{11} - C_{11} \D_{21} - C_{21}\end{pmatrix}\]Understanding matrix subtraction helps in simplifying complex expressions, just like in the given problem, where subtraction is used to achieve the final result matrix.

Elementary Linear Algebra

Elementary linear algebra is a branch of mathematics focused on the study of vectors, vector spaces, linear transformations, and systems of linear equations. It serves as the foundation for more advanced topics in algebra. Let's consider its relevance and application through the concept of matrices:

• Matrices can represent systems of linear equations, making them powerful tools in various mathematical and applied fields.
• Operations like addition and subtraction maintain the structure of linear equations, aiding in simplification and solution.
• Understanding these basic operations is crucial as they form the building blocks for more complex manipulations, such as finding determinants, eigenvalues, and transformations.

In practical scenarios, vectors and matrices are often used to model physical variables, compute transformations, and solve system equations. Thus, mastery of elementary linear algebra and its concepts like matrix operations offers a pathway to understanding higher analytical methods in mathematics and sciences.