Problem 18 - A Graphical Approach to College Algebra

Question

Perform each operation if possible. $$\left[\begin{array}{rr} 9 & 4 \\ -8 & 2 \end{array}\right]+\left[\begin{array}{rr} -3 & 2 \\ -4 & 7 \end{array}\right]$$

Step-by-Step Solution

Answer

The result of the matrix addition is \( \begin{bmatrix} 6 & 6 \\ -12 & 9 \end{bmatrix} \).

Step 1: Check matrix dimensions

Before performing matrix addition, we need to ensure both matrices have the same dimensions. Here, both matrices are 2x2 matrices, so they have the same dimensions, and addition is possible.

Step 2: Add corresponding elements

To add the matrices, add each element in the first matrix with the corresponding element in the second matrix. \[\begin{bmatrix} 9 & 4 \ -8 & 2 \end{bmatrix} + \begin{bmatrix} -3 & 2 \ -4 & 7 \end{bmatrix} = \begin{bmatrix} 9 + (-3) & 4 + 2 \ -8 + (-4) & 2 + 7 \end{bmatrix}\]

Step 3: Calculate element-wise sums

Calculate the sum for each corresponding element. \[\begin{bmatrix} 9 + (-3) & 4 + 2 \ -8 + (-4) & 2 + 7 \end{bmatrix} = \begin{bmatrix} 6 & 6 \ -12 & 9 \end{bmatrix}\]

Key Concepts

Matrix Operations

Matrix operations refer to a variety of mathematical techniques used to manipulate matrices. These operations include addition, subtraction, multiplication, and more advanced calculations like inverses and determinants. Each operation has its set of rules to follow.
- **Addition and Subtraction**: These are the most basic operations and require matrices to be of the same dimension. - **Multiplication**: This operation involves a bit more complexity as the number of columns in the first matrix must match the number of rows in the second. - **Scalar Multiplication**: Involves multiplying every element of a matrix by a constant number.
Understanding and mastering these operations can significantly help in solving systems of linear equations, performing transformations, and much more in various fields like physics and engineering.

2x2 Matrices

Matrices are rectangular arrays of numbers arranged in rows and columns. A 2x2 matrix specifically consists of two rows and two columns, forming a square matrix:\[\begin{bmatrix}a & b \c & d \end{bmatrix}\]
Each number or element in the matrix is denoted by its position. For example, in the matrix above, "a" is positioned in the first row, first column. 2x2 matrices are quite common and often used since they are simple yet provide a solid foundation for understanding more complex matrices.
They can be used to solve simple systems of linear equations, represent transformations in geometry, or model simple real-world scenarios. Working with 2x2 matrices often serves as the introduction to the world of linear algebra.

Element-wise Addition

Element-wise addition, also known as matrix addition, involves summing the corresponding entries of two matrices of the same dimensions. It is a straightforward process where each entry from one matrix is added to the corresponding entry in the other matrix. For example, if you have two 2x2 matrices:\[\begin{bmatrix}1 & 2 \3 & 4 \end{bmatrix}and \begin{bmatrix}5 & 6 \7 & 8 \end{bmatrix}\]
The resulting addition would be:\[\begin{bmatrix}1+5 & 2+6 \3+7 & 4+8 \end{bmatrix}= \begin{bmatrix}6 & 8 \10 & 12 \end{bmatrix}\]
This ensures each position in the matrices adds to create a new matrix of the same size. It is essential for students to ensure matrices are of identical dimensions before attempting element-wise addition, otherwise the operation is undefined.