Problem 13 - Algebra 1

Question

Find the sum of the matrices. $$ \left[\begin{array}{rrr} 1 & -2 & 2 \\ 0 & -3 & 4 \end{array}\right]+\left[\begin{array}{rrr} 3 & -4 & 5 \\ -8 & 1 & 6 \end{array}\right] $$

Step-by-Step Solution

Answer

The sum of the matrices is \[ \begin{array}{rrr} 1+3 & -2+(-4) & 2+5 \ 0+(-8) & -3+1 & 4+6 \end{array} \] which simplifies to \[ \begin{array}{rrr} 4 & -6 & 7 \ -8 & -2 & 10 \end{array} \].

Step 1: Identify the corresponding elements in the matrices

Since these are 2x3 matrices, it essentially means that they both have 2 rows and 3 columns. Therefore, the corresponding elements for these matrices will be at the same position in each of their corresponding rows and columns. For instance, the element '-2' in the first row and second column of the first matrix has its corresponding element '-4' in the first row and second column of the second matrix.

Step 2: Add the corresponding elements

Here we will add every pair of corresponding elements in each row and column position from the two matrices. For example, in the first row and second column, '-2' and '-4' are corresponding elements. The sum is '-2' + '-4' = '-6'. We simply need to apply the same process to every pair of corresponding elements in the two matrices.

Step 3: Display the resulting matrix

The resulting matrix will be found by placing the resulting sums from step 2 in their respective positions. The positions will follow the row and column of the corresponding elements that were added.

Key Concepts

2x3 matrices

Understanding the structure of a matrix is fundamental in mastering matrix addition and other operations. A 2x3 matrix is an array of numbers that consists of 2 rows and 3 columns. In other terms, it's like a grid with 2 horizontal lines (rows) and 3 vertical lines (columns), and every box created by the intersection of these lines is filled with a number. These matrices can represent various types of data in mathematics, physics, computer science, and more.

For instance, take the matrix: \[\left[\begin{array}{rrr} a & b & c \ d & e & f\end{array}\right]\] Here, 'a' through 'f' are elements that occupy this 2x3 space. When we're working with these matrices, whether adding, subtracting or performing other operations, the dimensions (in this case, 2x3) are crucial since they dictate the rules for the operations - like you can only add or subtract matrices of the same dimensions.

Corresponding elements in matrices

To successfully perform matrix addition, one must understand the concept of corresponding elements in matrices. Corresponding elements can be thought of as 'partners' or 'matches' located at the same spot within different matrices of the same size.

Consider two matrices, A and B. If they are both 2x3 matrices, then each element in matrix A has a precise counterpart in matrix B. The element located in the first row and the first column of matrix A corresponds to the element in the first row and the first column of matrix B, and so on. These pairs are what we refer to as corresponding elements. During addition, these elements are summed up individually to create a new matrix. For example, if the element in A is 4 and in B is 5, in the same position, their sum is 9, which will occupy the same position in the resulting matrix.

Adding matrices step by step

Adding matrices is a straightforward process, but meticulous attention to each step ensures accuracy. Here's a concise guide on adding matrices step by step:

Step 1: Check the Matrices' Dimensions

First, we must ensure that the two matrices are of the same size, meaning they have an equal number of rows and columns. Only matrices with the same dimensions can be added.

Step 2: Find Corresponding Elements

Locate the corresponding elements between the two matrices. These elements are in the exact same position in each matrix.

Step 3: Add the Elements

Simple addition is performed between each pair of corresponding elements. Do this for each element across the entire matrices.

Step 4: Create the Resultant Matrix

The sums computed in the previous step will occupy the same positions as the elements that were added to form the resulting matrix. At the end of this step, you'll have a new matrix that is the sum of the two original matrices.

By following these steps carefully, matrix addition should become a clear and manageable task. Remember, if at any point the dimensions do not match or the positions do not correspond, the addition cannot be performed and you must re-evaluate the matrices you are working with.