Problem 12 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression. $$ \left[\begin{array}{rr} 6 & 8 \\ -1 & 0 \end{array}\right]+\left[\begin{array}{rr} 0 & 5 \\ -3 & -1 \end{array}\right]+\left[\begin{array}{rr} -11 & -7 \\ 2 & -1 \end{array}\right] $$

Step-by-Step Solution

Answer

The result of the addition of the given matrices is \[\left[\begin{array}{rr} -5 & 6 \ -2 & -2 \end{array}\right]\]

Step 1: Identify the Corresponding Elements

Since two matrices can be added together only if they have the same dimensions, it's required that they are the same size in order to perform this operation. Here, all three matrices are 2x2 matrices, meaning you can indeed add them together. Like terms are in the same position in each matrix. Therefore, you will add together the terms that are in the same positions in each matrix.

Step 2: Add the Corresponding Elements

Add together each set of like terms to form a new matrix. Which yields:\[\left[\begin{array}{rr} 6 + 0 - 11 & 8 + 5 - 7 \ -1 - 3 + 2 & 0 - 1 - 1 \end{array}\right] \]

Step 3: Simplify the Matrix

Simplify each of the terms in the new matrix to give the final answer:\[\left[\begin{array}{rr} -5 & 6 \ -2 & -2 \end{array}\right] \]

Key Concepts

2x2 Matrices

When working with matrices, a common size that you will encounter is the 2x2 matrix. A 2x2 matrix has two rows and two columns, forming a small square block of four elements. These matrices are organized in a grid-like format:

\[ \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \ \end{bmatrix}\]
Here, each element in the matrix is denoted by a letter with two subscripts. The first subscript indicates the row number, and the second indicates the column number. This makes it easy to pinpoint each element's exact location.

• a₁₁ is the element in the first row, first column.
• a₁₂ is the element in the first row, second column.
• a₂₁ is the element in the second row, first column.
• a₂₂ is the element in the second row, second column.

These matrices are often used in various mathematical computations, including solving linear equations and representing transformations. Understanding how to handle them is a fundamental skill in matrix algebra.

Matrix Operations

Matrix operations are procedures we perform on matrices to manipulate or analyze them. One of the simplest and most common operations is matrix addition. Before we add any matrices, it is crucial to ensure they are of the same dimensions. In our example, all matrices are of dimension 2x2, which allows us to add them.

To perform matrix addition, we combine matrices by adding their corresponding elements. This is a straightforward but essential operation that helps in various applications, like calculating combined effects in systems or accumulations in data analysis.

• Ensure both matrices to be added are the same size.
• Identify corresponding elements in each matrix, maintaining the order of rows and columns.
• Add these corresponding elements to form a new matrix.

Matrix addition is associative and commutative, which means the order in which you add matrices does not matter. Learning to perform these operations serves as a stepping stone for more complex matrix calculations.

Element-wise Addition

Element-wise addition is a crucial concept in matrix operations. It refers to the process of adding corresponding elements from two or more matrices. Understanding this concept simplifies the process of matrix addition substantially. In our exercise, the three 2x2 matrices are added element-wise.

To perform element-wise addition:

• Start with the first element of each matrix (top-left corner), and add them.
• Proceed to the next element (top-right corner), and do the same.
• Move down to the second row and repeat this for each column.

For example, if we denote the matrices as \(A\), \(B\), and \(C\), their sum, matrix \(D\), would be:

\[ D = \begin{bmatrix} (A_{11} + B_{11} + C_{11}) & (A_{12} + B_{12} + C_{12}) \ (A_{21} + B_{21} + C_{21}) & (A_{22} + B_{22} + C_{22}) \end{bmatrix}\]
The result is a new matrix where each element is the sum of the elements found in the same position in each of the original matrices. This method ensures an easy and efficient way to handle matrix addition, making it a friendly operation to perform in various applications.