Problem 10 - College Algebra

Question

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined. $$ A=\left[\begin{array}{ll}{1} & {3} \\ {0} & {7}\end{array}\right], B=\left[\begin{array}{cc}{2} & {14} \\ {22} & {6}\end{array}\right], C=\left[\begin{array}{cc}{1} & {5} \\ {8} & {92} \\ {12} & {6}\end{array}\right], D=\left[\begin{array}{cc}{10} & {14} \\ {7} & {2} \\\ {5} & {61}\end{array}\right], E=\left[\begin{array}{cc}{6} & {12} \\ {14} & {5}\end{array}\right], F=\left[\begin{array}{cc}{0} & {9} \\ {78} & {17} \\\ {15} & {4}\end{array}\right] $$ $$ C+F $$

Step-by-Step Solution

Answer

The sum of \( C + F \) is \( \begin{bmatrix} 1 & 14 \\ 86 & 109 \\ 27 & 10 \end{bmatrix} \).

Step 1: Determine Matrix Dimensions

First, observe the dimensions of matrices \( C \) and \( F \). Matrix \( C \) is a \(3 \times 2\) matrix, meaning it has 3 rows and 2 columns. Similarly, matrix \( F \) is also a \(3 \times 2\) matrix with 3 rows and 2 columns.

Step 2: Evaluate Compatibility for Addition

In order to perform matrix addition, the matrices must have the same dimensions. Since both matrices \( C \) and \( F \) are of dimensions \(3 \times 2\), they can indeed be added.

Step 3: Perform the Addition

Add corresponding elements of matrices \( C \) and \( F \). This means:- \( (1+0), (5+9) \) for the first row,- \( (8+78), (92+17) \) for the second row,- \( (12+15), (6+4) \) for the third row.

Step 4: Compute the Result

Calculate each element in the resulting matrix:- First row: \( 1+0 = 1 \), \( 5+9 = 14 \)- Second row: \( 8+78 = 86 \), \( 92+17 = 109 \)- Third row: \( 12+15 = 27 \), \( 6+4 = 10 \)The result of the addition is the matrix:\[\left[\begin{array}{cc}1 & 14 \ 86 & 109 \ 27 & 10\end{array}\right]\]

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is the first step in working with matrix operations. Each matrix is viewed as a rectangular array of numbers, and its dimensions are defined by the number of rows and columns it has. For example, if a matrix has 3 rows and 2 columns, it is described as a \(3 \times 2\) matrix. This notation helps in categorizing matrices by size, and it's crucial for determining which operations are feasible. When working on exercises like adding or subtracting matrices, always list the dimensions first. It's like checking the size of a puzzle piece before connecting it with others. If matrices do not have the expected sizes, some operations might be impossible to perform. For instance, in this exercise, both matrix \(C\) and matrix \(F\) have dimensions \(3 \times 2\), making it possible to proceed with the matrix addition.

Matrix Compatibility

Matrix compatibility is essential when performing operations such as addition or subtraction. To add two matrices together, they must be of the same dimensions. This means they must have the same number of rows and columns. This requirement ensures that each element in one matrix has a corresponding element in the other for the arithmetic operation. For example, in our exercise, matrices \(C\) and \(F\) each have three rows and two columns. This parallel structure allows us to add them element-wise. Checking for compatibility is as crucial as checking ingredients for a recipe – the right elements must be present in the right quantities.

Matrix Operations

Matrix operations involve applying arithmetic processes to each element of the matrices involved. In the context of addition, as illustrated in the exercise, the operation consists of adding the corresponding elements from two matrices to produce their sum. Let's break down this step further to make it clear. When adding matrices \(C\) and \(F\), each element in the same position from both matrices is added together:

• First row: add \(1+0\), \(5+9\)
• Second row: add \(8+78\), \(92+17\)
• Third row: add \(12+15\), \(6+4\)

The resulting matrix will consist of these summed elements properly aligned, resulting in a new matrix from the operation. Understanding and practicing these operations can simplify and speed up computations in various scientific and engineering tasks.