Problem 32 - College Algebra

Question

Use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) CA

Step-by-Step Solution

Answer

The result of \(CA\) is \( \begin{bmatrix} 10 & -20 \\ -5 & -25 \\ -10 & 20 \end{bmatrix} \).

Step 1: Determine Operation Type

The given operation is the multiplication of matrix \(C\) by matrix \(A\), denoted as \(CA\). Matrix multiplication involves multiplying corresponding entries and summing the results.

Step 2: Check Dimensions for Multiplication

Matrix \(C\) is a \(3 \times 2\) matrix and matrix \(A\) is a \(2 \times 2\) matrix. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix. Here, \(C\) has 2 columns and \(A\) has 2 rows, so the multiplication is possible.

Step 3: Calculate CA

To calculate \(CA\), multiply each row of \(C\) by each column of \(A\) and sum the results. The resulting matrix will have the same number of rows as \(C\) and the same number of columns as \(A\), resulting in a \(3 \times 2\) matrix. 1. First row of \(C\) times each column of \(A\):- Row 1: \([-1( -10) + 0(5), -1(20) + 0(25)] = [10, -20]\)2. Second row of \(C\) times each column of \(A\):- Row 2: \([0( -10) + (-1)(5), 0(20) + (-1)(25)] = [-5, -25]\)3. Third row of \(C\) times each column of \(A\):- Row 3: \([1( -10) + 0(5), 1(20) + 0(25)] = [-10, 20]\)The resulting \(CA\) matrix is \( \begin{bmatrix} 10 & -20 \ -5 & -25 \ -10 & 20 \end{bmatrix} \).

Key Concepts

Understanding Matrix Dimensions

When we talk about matrix dimensions, we are essentially describing the shape and size of a matrix. Each matrix has dimensions given in terms of rows and columns. For example, if a matrix has 3 rows and 2 columns, we say it is a \(3 \times 2\) matrix. These dimensions are crucial, especially when considering matrix operations.

When two matrices are involved in an operation like multiplication, their compatibility is determined by their dimensions. In the original exercise, we dealt with matrices \(C\) and \(A\). Here, matrix \(C\) is a \(3 \times 2\) matrix and matrix \(A\) is a \(2 \times 2\) matrix. Ensure to always check that the number of columns in the first matrix matches the number of rows in the second one.

Why is this check important? It ensures each element of the row in the first matrix has a corresponding element in the column of the second matrix to pair with in the multiplication process. This dimensional rule helps us proceed to correct calculation in matrix operations without errors.

Basics of Matrix Operations

Matrix operations include addition, subtraction, and multiplication, with each having specific rules and requirements. In this exercise, our focus is on matrix multiplication. Unlike simple arithmetic multiplication, matrix multiplication is a process of combining entries in rows from one matrix with entries in columns of another matrix.

This might sound intimidating, but it's simpler once you break it down to these steps:

• Identify compatible matrices where the columns of the first match the rows of the second.
• Multiply each element of a row in the first matrix with the corresponding element in a column of the second matrix.
• Sum the products to get a single number in the resulting matrix.

Matrix \(CA\) from our problem is achieved by repeatedly applying this technique over each row of \(C\) and each column of \(A\). Once you understand and practice these steps, the process becomes routine.

Educational Problem Solving Strategies

Solving educational problems such as matrix multiplication requires a blend of patience and practice. Start by clearly understanding the concept. Conditioning your mind to follow structured steps makes tackling complex problems simpler.

To enhance your problem-solving skills, adopt these strategies:

• Always start by analyzing the problem carefully. Determine what is required from the problem statement.
• Break down the process into smaller, manageable steps as shown in the step-by-step solution above.
• Practicing on various matrix problems will reinforce these concepts, making them second nature.

Remember, there’s no rush in problem-solving. Understanding the basics will allow you to solve similar problems with ease in the future. This method also empowers you to explain your solutions to others—a sure sign of mastery.