Problem 35 - Algebra 2

Question

Perform the indicated matrix operation. If the matrix does not exist, write impossible. $$ \left[\begin{array}{cc}{1.35} & {5.80} \\ {1.24} & {14.32} \\ {6.10} & {35.26}\end{array}\right]+\left[\begin{array}{cc}{0.45} & {3.28} \\ {1.94} & {16.72} \\ {4.31} & {21.30}\end{array}\right] $$

Step-by-Step Solution

Answer

The sum of the matrices is \( \begin{bmatrix} 1.80 & 9.08 \\ 3.18 & 31.04 \\ 10.41 & 56.56 \end{bmatrix} \).

Step 1: Check Matrix Dimensions

Before performing any operations, we need to check the dimensions of the matrices. Both matrices given are 3x2 matrices, which means they each have 3 rows and 2 columns. Since they have the same dimensions, we can proceed with the addition.

Step 2: Perform Matrix Addition

To find the sum of two matrices, you add corresponding elements. This means you will add the element in the first row and first column of the first matrix to the element in the first row and first column of the second matrix, and so on. The operation goes as follows:\[\text{Sum} = \begin{bmatrix} 1.35 + 0.45 & 5.80 + 3.28 \ 1.24 + 1.94 & 14.32 + 16.72 \ 6.10 + 4.31 & 35.26 + 21.30 \end{bmatrix}\]

Step 3: Calculate Each Element

We perform the addition for each pair of corresponding elements in the matrices:- \(1.35 + 0.45 = 1.80\)- \(5.80 + 3.28 = 9.08\)- \(1.24 + 1.94 = 3.18\)- \(14.32 + 16.72 = 31.04\)- \(6.10 + 4.31 = 10.41\)- \(35.26 + 21.30 = 56.56\)

Step 4: Write the Resulting Matrix

After calculating each element, we write the resulting matrix:\[\begin{bmatrix} 1.80 & 9.08 \ 3.18 & 31.04 \ 10.41 & 56.56 \end{bmatrix}\]

Key Concepts

Matrix Dimensions

Matrix dimensions are vital to understand before performing any matrix operations. Each matrix has dimensions that tell us the number of rows and columns it contains. Typically, dimensions are written in the form \( m \times n \), where \( m \) is the number of rows, and \( n \) is the number of columns. For the given matrices, both are 3x2, meaning each has 3 rows and 2 columns. This consistency is crucial because, in order to perform addition, two matrices must have identical dimensions. This ensures that there is a one-to-one correspondence between their elements, allowing the operation to proceed smoothly.

Corresponding Elements

The concept of corresponding elements is central to matrix addition. When adding matrices, you combine elements that are in the same position in each matrix.
For example, if you have two matrices, the element located at the first row, first column of the first matrix is added to the element at the same position in the second matrix.

• This means that each element in a row and column of one matrix has a partner in the other matrix.
• Only these similarly positioned elements are combined.

Understanding this alignment is crucial since it ensures that each sum you calculate relates to a specific position in the resulting matrix.

Matrix Operation

Matrix operation, particularly addition, involves executing arithmetic with corresponding numbers in matrices that share the same dimensions. The operations extend various simple arithmetic principles to two-dimensional structures.

• Only matrices of the same dimensions can be added together.
• The result of a matrix addition is a new matrix of the same dimension.

In the exercise at hand, you are adding each element from the first matrix to its corresponding element in the second: \( 1.35 + 0.45 \), \( 5.80 + 3.28 \), and so forth. Each resulting number occupies the same row and column position as in the original matrices, leading to a new matrix product.

3x2 Matrices

The term 3x2 matrices describes matrices with three rows and two columns. These matrices play a fundamental role in numerous mathematical applications, providing a compact way to handle and organize data.
For the 3x2 matrices you are working with:

• Each row corresponds to a separate equation or data set.
• Each column signifies a particular variable or category within those equations or data sets.

These structures are particularly useful in scenarios like data representation or in the case of our problem, simplifying operations such as addition. Due to their manageable size, these matrices serve as excellent examples for beginners learning about matrix operations, like the addition shown in this exercise.