Problem 17 - A Graphical Approach to College Algebra

Question

Perform each operation if possible. $$\left[\begin{array}{rrr} 6 & -9 & 2 \\ 4 & 1 & 3 \end{array}\right]+\left[\begin{array}{rrr} -8 & 2 & 5 \\ 6 & -3 & 4 \end{array}\right]$$

Step-by-Step Solution

Answer

The result is \(\begin{bmatrix} -2 & -7 & 7 \\ 10 & -2 & 7 \end{bmatrix}.\)

Step 1: Understand Matrix Addition

Matrix addition involves adding corresponding elements in each matrix. To add two matrices, they must have the same dimensions. Here, both matrices are 2x3, which means adding is possible.

Step 2: Add Corresponding Elements in the First Row

Add the elements of the first row from each matrix:\[\begin{bmatrix} 6 & -9 & 2 \end{bmatrix} + \begin{bmatrix} -8 & 2 & 5 \end{bmatrix} = \begin{bmatrix} 6 + (-8) & -9 + 2 & 2 + 5 \end{bmatrix} = \begin{bmatrix} -2 & -7 & 7 \end{bmatrix}\]

Step 3: Add Corresponding Elements in the Second Row

Next, add the elements of the second row from each matrix:\[\begin{bmatrix} 4 & 1 & 3 \end{bmatrix} + \begin{bmatrix} 6 & -3 & 4 \end{bmatrix} = \begin{bmatrix} 4 + 6 & 1 + (-3) & 3 + 4 \end{bmatrix} = \begin{bmatrix} 10 & -2 & 7 \end{bmatrix}\]

Step 4: Combine Results

Combine the results from each row to form the resulting matrix:\[\begin{bmatrix} -2 & -7 & 7 \ 10 & -2 & 7 \end{bmatrix}\]

Key Concepts

Matrix Operations

Matrix operations are fundamental in linear algebra and involve various methods like addition, subtraction, and multiplication of matrices. In this exercise, we focus on matrix addition, which is one of the simplest yet essential matrix operations. To perform matrix addition, you must ensure that both matrices have the same dimensions. This means the number of rows and columns must match in each matrix.
A matrix is essentially a rectangular array of numbers arranged in rows and columns. In our case, each matrix has dimensions of 2x3, which means 2 rows and 3 columns. Only matrices of the same size can be added directly, where corresponding elements are combined. For instance:

• The element from the first row and first column of both matrices are added together.
• This process is repeated for each corresponding element across the whole matrix.

Matrix addition is both commutative and associative, meaning the order in which matrices are added does not change the outcome. So, if you have matrices A and B, then A + B = B + A.

Algebra

In the context of matrices, algebra helps us manage and manipulate complex relationships among numbers. Algebra allows us to apply rules and methods to perform operations effectively. The same basic algebraic principles apply when working with matrices as they do with numbers. For example:

• When adding matrices, we apply the rule of adding like terms, which means adding numbers located in the same position within each matrix.
• Negative numbers are treated according to standard algebraic rules, such as adding a negative number is equivalent to subtraction.

Through these algebraic rules, the process of matrix addition becomes straightforward, as shown in the steps of breaking down the operation by each row. Algebra provides the structure and logic required to handle matrices systematically.

Elementary Algebra

Elementary algebra forms the groundwork for understanding more advanced algebraic concepts, including those involving matrices. It introduces basic operations like addition and subtraction, which are applied similarly when working with matrices in linear algebra.
Here, each step of matrix addition can be seen as an application of elementary algebra's foundation:

• Comprehending signed numbers and order of operations is crucial to correctly adding each pair of elements.
• Recognizing equivalent expressions helps simplify calculations, ensuring that the same approach used in basic algebra for single numbers can scale up when dealing with matrices.

Thus, understanding elementary algebra is fundamental, as it equips students with the skills needed to perform operations like matrix addition with accuracy and confidence. By drawing from these basic principles, solving matrix problems becomes an accessible task.