Problem 40 - Precalculus: Mathematics for Calculus

Question

The matrices \(A, B,\) and \(C\) are defined as follows. $$A=\left[\begin{array}{rrr}0.3 & 1.1 & 2.4 \\\0.9 & -0.1 & 0.4 \\\\-0.7 & 0.3 & -0.5\end{array}\right] \quad B=\left[\begin{array}{cc}1.2 & -0.1 \\\0 & -0.5 \\\0.5 & -2.1\end{array}\right]$$ $$C=\left[\begin{array}{rrr}-0.2 & 0.2 & 0.1 \\\1.1 & 2.1 & -2.1\end{array}\right]$$ Use a graphing calculator to carry out the indicated algebraic operation, or explain why it cannot be performed. $$C B$$

Step-by-Step Solution

Answer

The matrix product \( CB \) is \( \left[ \begin{array}{cc} -0.19 & -0.29 \\ 0.27 & 3.25 \end{array} \right] \).

Step 1: Determine Matrix Dimensions

First, identify the dimensions of matrices \( C \) and \( B \). Matrix \( C \) is a \( 2 \times 3 \) matrix, and Matrix \( B \) is a \( 3 \times 2 \) matrix.

Step 2: Assess Multiplication Compatibility

For matrix multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, \( C \) has 3 columns, and \( B \) has 3 rows, so they are compatible for multiplication.

Step 3: Find the Resulting Matrix Dimensions

The product matrix \( CB \) will have the dimensions based on the rows of \( C \) and the columns of \( B \), resulting in a \( 2 \times 2 \) matrix.

Step 4: Multiply the Matrices

Perform the matrix multiplication:- For the element in the first row and first column of \( CB \), calculate \((-0.2)(1.2) + (0.2)(0) + (0.1)(0.5) = -0.24 + 0 + 0.05 = -0.19\).- For the element in the first row and second column of \( CB \), compute \((-0.2)(-0.1) + (0.2)(-0.5) + (0.1)(-2.1) = 0.02 - 0.1 - 0.21 = -0.29\).- For the element in the second row and first column of \( CB \), calculate \((1.1)(1.2) + (2.1)(0) + (-2.1)(0.5) = 1.32 + 0 - 1.05 = 0.27\).- For the element in the second row and second column of \( CB \), calculate \((1.1)(-0.1) + (2.1)(-0.5) + (-2.1)(-2.1) = -0.11 - 1.05 + 4.41 = 3.25\).

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial in any matrix-related operation, such as matrix multiplication. The dimensions of a matrix are labeled as "rows x columns." This means, if a matrix has 2 rows and 3 columns, it is a "2 x 3" matrix. Knowing the dimensions helps in identifying how the matrix looks and also in determining its compatibility for operations with other matrices. For instance, Matrix \( C \) is a \( 2 \times 3 \) matrix and Matrix \( B \) is a \( 3 \times 2 \) matrix.

Here’s how to accurately identify the matrix dimensions:

• Count the number of horizontal lines of elements—these are the "rows."
• Count the number of vertical lines of elements—these are the "columns."

By learning how to determine matrix dimensions quickly, it becomes easier to recognize the formation and layout of any matrix, making further operations much smoother.

Matrix Compatibility

Matrix compatibility is an essential concept when performing matrix multiplication. It's all about whether two matrices can be multiplied together. The rule of thumb is simple: the number of columns in the first matrix must match the number of rows in the second matrix.

For example, consider matrices \( C \) and \( B \):

• Matrix \( C \) ([[2, 3]]) has 3 columns.
• Matrix \( B \) ([[3, 2]]) has 3 rows.

Since the number of columns in \( C \) aligns with the number of rows in \( B \), these two matrices are compatible for multiplication.

If the dimensions didn't match up, multiplication wouldn't be feasible. Understanding this compatibility is key to executing matrix operations without errors and achieving correct results.

Graphing Calculator

A graphing calculator can be a powerful tool for performing matrix operations, especially matrix multiplication. It simplifies the process by allowing users to input the matrices, set the multiplication operation, and receive the result with minimal manual computation. Here’s how you can effectively utilize a graphing calculator:

• Begin by entering each matrix into the calculator. Use the calculator's "matrix" function to do this.
• Select the operation (e.g., multiplication). Ensure the matrices are in the correct order.
• The calculator will compute the result, showing you the product matrix.

This process can save time and reduce errors. It's a helpful resource for students who might find manual calculations daunting. Plus, it’s a great way to verify the results of manual calculations, ensuring accuracy and consistency in complex linear algebra tasks.

Linear Algebra

Linear algebra is a branch of mathematics focused on vector spaces and the linear mapping between them. It includes the study of matrices, systems of linear equations, determinants, and vector spaces.

When dealing with matrices and matrix multiplication, we are rooted in the principles of linear algebra. This area of mathematics is essential because:

• It forms the foundation for more complex areas in mathematics and applied sciences.
• Matrix operations are used to solve systems of equations, which is a key application of linear algebra.
• Understanding linear transformations and their representations through matrices can solve many geometric and computational problems.

Linear algebra is a powerful tool that supports a range of practical applications, from computer graphics to scientific computing. By mastering its concepts, students open doors to a deeper understanding of the mathematical world and its real-world applications.