Problem 15 - Algebra 2

Question

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ \left[\begin{array}{rr}{-11} & {4} \\ {-3} & {6}\end{array}\right]+\left[\begin{array}{rr}{-2} & {-5} \\ {5} & {-3}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix is \(\begin{bmatrix} -13 & -1 \\ 2 & 3 \end{bmatrix}\).

Step 1: Understand Matrix Addition

Matrix addition is only possible when both matrices have the same dimensions. Here, both matrices have the dimensions 2x2, meaning they can be added together.

Step 2: Adding Corresponding Elements

To add two matrices, add their corresponding elements. Take each element from the same position of the two matrices and add them together. Calculate as follows:- Top left: \( -11 + (-2) = -13 \)- Top right: \( 4 + (-5) = -1 \)- Bottom left: \( -3 + 5 = 2 \)- Bottom right: \( 6 + (-3) = 3 \)

Step 3: Write the Resulting Matrix

Combine the sums from Step 2 into a single matrix. The result of adding the two matrices is:\[\begin{bmatrix}-13 & -1 \2 & 3\end{bmatrix}\]

Key Concepts

Matrix Operations

Matrix operations are fundamental in linear algebra and involve various techniques to manipulate matrices. These operations include addition, subtraction, multiplication, and sometimes more advanced techniques like finding an inverse. In this exercise, we focus on matrix addition.
When performing matrix additions, such as adding two 2x2 matrices, it's crucial that each matrix has matching dimensions. If they do not, addition is impossible. Here, both matrices are 2x2, ensuring that they can indeed be added.

• The first rule of matrix addition is that it is only defined for matrices of the same size.
• Matrix addition involves adding each element from one matrix to the corresponding element in the other matrix.
• This element-wise addition results in a new matrix of the same dimensions.

By understanding these principles, one can effectively solve any basic matrix addition problem. Remember, the operations must follow the order of their positions in the matrices.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix in linear algebra. It has two rows and two columns, which presents four elements in total. Understanding the layout of a 2x2 matrix helps greatly in performing operations like addition.
Here is a typical structure of a 2x2 matrix:

• The top-left element, labeled as a11.
• The top-right element, labeled as a12.
• The bottom-left element, labeled as a21.
• The bottom-right element, labeled as a22.

Using this reference, when performing operations, each of these positions can be visualized separately, making calculations straightforward. For instance, while adding two matrices, one would add the a11 element of the first matrix to the a11 element of the second, and so on through the matrix.

Elements of a Matrix

The elements of a matrix are the individual numbers that populate it. They are located at specific positions in each matrix, which are critical for matrix operations.
When two matrices are added, it's these elements that are combined. Every element has a specific notation based on its row and column position, which helps identify and organize the calculations required.

• In a matrix, the horizontal components are called rows, and the vertical components are called columns.
• The position of an element is given by \(a_{ij}\), where *i* is the row number and *j* is the column number.
• In matrix addition, elements are added by focusing on their \(a_{11}\) to \(a_{22}\) correspondence.

Knowing the position and value of each element is key to successfully carrying out matrix operations, ensuring accuracy and consistency in your results. This knowledge is the foundation for more complex matrix manipulations.