Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Perform each operation if possible. $$\left[\begin{array}{rr}6 & -2 \\ 5 & 4\end{array}\right]+\left[\begin{array}{rr}-1 & 7 \\ 7 & -4\end{array}\right]$$

Step-by-Step Solution

Answer

The result is \( \begin{array}{cc} 5 & 5 \\ 12 & 0 \end{array} \).

Step 1: Verify Matrix Dimensions

Check if the two matrices have the same dimensions. Both matrices are 2x2, meaning they each have 2 rows and 2 columns. Since they are of the same dimensions, addition is possible.

Step 2: Add Corresponding Elements

Add the elements located in the same positions in the two matrices: \[ \begin{array}{cc} (6 + (-1)) & (-2 + 7) \ (5 + 7) & (4 + (-4))\end{array} \] This results in the following matrix: \[ \begin{array}{cc} 5 & 5 \ 12 & 0\end{array}\]

Step 3: Write the Resulting Matrix

The resulting matrix after performing the addition is: \[ \begin{array}{cc} 5 & 5 \ 12 & 0\end{array} \]

Key Concepts

Matrix Operations

Matrix operations are tools that help us manipulate matrices to solve various mathematical problems. One of the basic operations in matrix algebra is addition. To add two matrices, they must have the same dimensions. This means that each matrix should have the same number of rows and columns. This requirement ensures that each element in one matrix has a corresponding element in the other matrix.
Matrix addition involves adding each pair of corresponding elements. For instance, if we have two 2x2 matrices, the element in the first row and first column of the first matrix is added to the element in the first row and first column of the second matrix, and so on.
This operation is straightforward:

• First, ensure both matrices have the same size.
• Add each element in the first matrix to the corresponding element in the second matrix.
• The result is a new matrix with the same dimensions.

By following these steps, you can easily perform matrix addition.

Linear Algebra

Linear algebra is a branch of mathematics that focuses on vectors, vector spaces, and linear mappings between these spaces. It also looks into systems of linear equations and their solutions.
Matrices are essential components of linear algebra and are used to represent and solve linear equations. By understanding matrix operations, like addition, one can begin to perform more complex tasks such as matrix multiplication, finding determinants, and inverses, which are pivotal in solving linear equations.
Linear algebra finds applications in various fields including:

• Computer Science: For graphics and machine learning.
• Physics: To describe physical phenomena.
• Economics: For modeling and solving systems of equations.

Starting with simple operations like matrix addition, students can gradually build their skills in linear algebra.

2x2 Matrices

The 2x2 matrix is a fundamental building block in the study of matrices. This type of matrix consists of two rows and two columns, forming a grid of four elements. Mathematically, it can be represented in the form: \[ \begin{array}{cc} a & b \c & d \\end{array}\]These matrices are easy to visualize and manipulate, making them a great starting point for learning matrix operations.In the context of addition, working with 2x2 matrices is often an introductory exercise because of their simplicity and ease of calculation.Some properties of 2x2 matrices include:

• Determinant: Important for finding matrix inverses.
• Trace: The sum of the elements on the main diagonal (a + d).
• Matrix multiplication: A useful operation for combining transformations.

Through practice, students become familiar with these operations, which serve as a foundation for more advanced concepts in linear algebra.