Problem 15 - Algebra 1

Question

Find the sum of the matrices. $$ \left[\begin{array}{rr} 6.2 & -1.2 \\ -2.5 & -4.4 \\ 3.4 & -5.8 \end{array}\right]+\left[\begin{array}{rr} 1.5 & 9.2 \\ 6.6 & -2.2 \\ 5.7 & -7.1 \end{array}\right] $$

Step-by-Step Solution

Answer

The sum of the two matrices is:\[\begin{array}{r r}7.7 & 8 \4.1 & -6.6 \9.1 & -12.9 \end{array}\]

Step 1: Verify matrix dimensions

Examine both matrices to ensure they have the same dimensions. Here, both matrices are of size 3x2, so they can be added together.

Step 2: Add corresponding entries

For each element in the same row and column position in both matrices, perform the addition operation. \[\begin{array}{r r}6.2 + 1.5 & -1.2 + 9.2 \-2.5 + 6.6 & -4.4 - 2.2 \3.4 + 5.7 & -5.8 - 7.1 \end{array}\]

Step 3: Formulate the result

The result matrix is the sum of each corresponding element in the two matrices.\[\begin{array}{r r}7.7 & 8 \4.1 & -6.6 \9.1 & -12.9 \end{array}\]

Key Concepts

Introduction to Matrices

Matrices are arrangements of numbers, symbols, or expressions, **organized in rows and columns**. They are a fundamental concept in mathematics, often used to solve problems in linear algebra. When dealing with matrices, you might represent them using brackets or parentheses. For example,

• A 2x2 matrix has two rows and two columns.
• A 3x2 matrix has three rows and two columns.

This format makes it easier to manage data and apply a variety of operations, such as addition or multiplication. One important aspect of matrices is that they can be added together only if they have the same dimensions, ensuring that each element aligns perfectly with its counterpart in the other matrix.

Basics of Algebra with Matrices

Algebra involving matrices is very similar to regular algebra, except that instead of adding or multiplying numbers, you perform these operations on entire **rows and columns**. In addition to basic operations like addition and subtraction, matrices can also undergo more complex operations like multiplication and inversion. When you're adding two matrices, it's crucial to add only corresponding elements—those in the same row and column position.

• For example, if you have two matrices of dimension 3x2, adding them involves pairing the first element of the first row from each matrix, then the second element of the first row, and so on.
• Formulating the sum of matrices is similar to regular addition, but repeated for each element across all rows and columns.

This operation makes matrix algebra a powerful tool for solving systems of equations or optimizing linear functions.

Understanding Matrix Dimensions

Dimensions are a key feature of matrices, dictating how many **rows and columns** they have. The dimensions reveal much about what operations can be performed.

• A matrix with 3x2 dimensions means it has three rows and two columns.
• This also tells you immediately how these matrices can be added, multiplied, or even transformed through various mathematical procedures.

When adding two matrices, as in the given exercise, it's crucial first to confirm that their dimensions match. This step ensures that each element has a corresponding element in the other matrix, allowing you to perform straightforward addition for each individual element.

Relevance in Mathematics Education

Matrix addition is foundational in mathematics education as it provides insights into more complex mathematical concepts. Learning how to manipulate matrices is not just a theoretical exercise; it paves the way for practical applications.

• Students often encounter matrices when solving systems of equations, particularly in subjects like physics and engineering.
• Moreover, understanding the principles of matrices prepares students for higher education topics like computer graphics, data science, and machine learning.

By studying matrices, students develop a solid base in recognizing patterns and logical structuring. This knowledge is applicable across a wide variety of fields, making it a valuable component of comprehensive mathematical education.