Problem 26 - Precalculus with Limits

Question

In Exercises 25-28, use the matrix capabilities of a graphing utility to evaluate the expression. Round your results to three decimal places, if necessary. \(55\left(\left[\begin{array}{r} 14 & -11 \\ -22 & 19 \end{array}\right] + \left[\begin{array}{r} -22 & 20 \\ 13 & 6 \end{array}\right]\right)\)

Step-by-Step Solution

Answer

The evaluated expression results in the matrix \[ \left[\begin{array}{r} -440 & 495 \ -495 & 1375 \end{array}\right]\]

Step 1: Matrix Addition

Begin by adding the two matrices. This is done element by element. Add the corresponding elements of the two matrices. \[ \left[\begin{array}{r} 14 & -11 \ -22 & 19 \end{array}\right] + \left[\begin{array}{r} -22 & 20 \ 13 & 6 \end{array}\right] = \left[\begin{array}{r} 14 + -22 & -11 + 20 \ -22 + 13 & 19 + 6 \end{array}\right] = \left[\begin{array}{r} -8 & 9 \ -9 & 25 \end{array}\right]\]

Step 2: Scalar Multiplication

Next, multiply the resultant matrix from the previous step with the scalar 55. This is also performed element by element. Each element in the matrix is multiplied with the scalar. \[55 * \left[\begin{array}{r} -8 & 9 \ -9 & 25 \end{array}\right] = \left[\begin{array}{r} 55 * -8 & 55 * 9 \ 55 * -9 & 55 * 25 \end{array}\right] = \left[\begin{array}{r} -440 & 495 \ -495 & 1375 \end{array}\right]\]

Key Concepts

Matrix Addition

Understanding matrix addition is essential when working with matrices in algebra and precalculus. Think of a matrix as an array of numbers arranged in rows and columns.

When adding two matrices, you simply add the corresponding elements from each matrix together. This means that the position of each element matters; you only add elements that occupy the same position in each matrix.

To visualize this, imagine two matrices, A and B, where you add A's first row, first column element to B's first row, first column element, and so on for each position in the matrices. Here's a basic example:

• If matrix A has an element of 2 in the top-left corner and matrix B has a 3 in the same spot, their sum in that position will be 5.

However, remember that you can only add matrices of the same dimensions. If they have a different number of rows or columns, the addition is undefined.

Scalar Multiplication

Scalar multiplication involves taking a matrix and multiplying every element within that matrix by a single number, which is known as a 'scalar.'

This operation doesn't change the dimension of the original matrix, but it does scale each element. For instance, if every element in a matrix is multiplied by 2, the entire matrix becomes twice as large.

An important application of scalar multiplication is in the concept of dilation in geometry, where you can change the size of a shape without altering its properties by multiplying the coordinates by a scalar. Scalar multiplication is also foundational in operations involving matrix equations and transformations.

Graphing Utilities for Matrices

Graphing utilities come in handy for performing complex matrix operations without manual calculations. They are often integrated into graphing calculators or available as software on computers.

These utilities allow you to input matrices and automatically perform operations like addition, scalar multiplication, and even more advanced operations such as finding inverses and determinants.

When using graphing utilities, it's crucial to input the matrix correctly, respecting rows and columns, to ensure the operation is carried out accurately. Many educational environments now utilize software like MATLAB or GeoGebra, which include these graphing utilities for matrices, helping to visualize concepts in linear algebra and analytic geometry.

Rounding Decimal Places

When working with matrices in mathematical calculations, especially in precalculus, it's important to present answers in a clear and concise manner. Rounding to a certain number of decimal places is a method of achieving this clarity.

Be consistent in rounding across your work to maintain accuracy throughout your calculations. If a problem specifies rounding to three decimal places, for example, look at the fourth decimal place to decide whether to round up or down.

This is significant in real-world applications where precision matters, such as in engineering or finance. Still, even when the context is purely academic, proper rounding is a must for conveying the correct level of precision intended in the solution.