Problem 25 - 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. \(\dfrac{3}{7}\left[\begin{array}{r} 2 & 5 \\ -1 & -4 \end{array}\right] + 6\left[\begin{array}{r} -3 & 0 \\ 2 & 2 \end{array}\right]\)

Step-by-Step Solution

Answer

The result of the expression is \( \left[\begin{array}{r} -17.143 & 2.142 \ 11.572 & 10.286 \end{array}\right] \)

Step 1: Perform the Scalar Multiplication

Perform Scalar multiplications on both matrices. For scalar multiplication, each entry in the matrix is multiplied by the scalar. \(\dfrac{3}{7}\left[\begin{array}{r} 2 & 5 \ -1 & -4 \end{array}\right] = \left[\begin{array}{r} \dfrac{3}{7}*2 & \dfrac{3}{7}*5 \ \dfrac{3}{7}*(-1) & \dfrac{3}{7}*(-4) \end{array}\right] = \left[\begin{array}{r} 0.857 & 2.142 \ -0.428 & -1.714 \end{array}\right]\)\(6\left[\begin{array}{r} -3 & 0 \ 2 & 2 \end{array}\right] = \left[\begin{array}{r} 6*(-3) & 6*0 \ 6*2 & 6*2 \end{array}\right] = \left[\begin{array}{r} -18 & 0 \ 12 & 12 \end{array}\right]\)

Step 2: Perform the Matrix Addition

Perform the addition of the two resulting matrices from step 1. For matrix addition, corresponding elements in the matrix are added together.\(\left[\begin{array}{r} 0.857 & 2.142 \ -0.428 & -1.714 \end{array}\right] + \left[\begin{array}{r} -18 & 0 \ 12 & 12 \end{array}\right] = \left[\begin{array}{r} 0.857-18 & 2.142+0 \ -0.428+12 & -1.714+12 \end{array}\right] = \left[\begin{array}{r} -17.143 & 2.142 \ 11.572 & 10.286 \end{array}\right]\)

Key Concepts

Understanding Scalar Multiplication

Scalar multiplication is a fundamental operation in matrix algebra that involves multiplying every entry within a matrix by a scalar, which is just a fancy term for a real number. Let's break that down with an example. Suppose you have a matrix:

• Matrix: \[\begin{bmatrix} 2 & 5 \ -1 & -4 \end{bmatrix}\]

If we multiply this entire matrix by a scalar, say \(\frac{3}{7}\), we would individually multiply each element of the matrix by \(\frac{3}{7}\):

• \(\frac{3}{7} \times 2\) becomes approximately \(0.857\)
• \(\frac{3}{7} \times 5\) becomes approximately \(2.142\)
• \(\frac{3}{7} \times -1\) becomes approximately \(-0.428\)
• \(\frac{3}{7} \times -4\) becomes approximately \(-1.714\)

The resulting matrix \[\begin{bmatrix} 0.857 & 2.142 \ -0.428 & -1.714 \end{bmatrix}\]shows how each entry scales by the same factor. This operation is useful when adjusting the size or scale of data, and is commonly used in various applications involving computer graphics and data transformations.

Exploring Matrix Addition

Matrix addition is another basic yet crucial operation in linear algebra, which allows for the combination of two matrices of the same dimensions by adding corresponding elements. For successful addition, both matrices must have identical numbers of rows and columns.
Let's explore with the matrices already processed through scalar multiplication:

• Matrix 1: \[\begin{bmatrix} 0.857 & 2.142 \ -0.428 & -1.714 \end{bmatrix}\]
• Matrix 2: \[\begin{bmatrix} -18 & 0 \ 12 & 12 \end{bmatrix}\]

To add these matrices, you add up corresponding elements:

• Top left: \(0.857 + (-18) = -17.143\)
• Top right: \(2.142 + 0 = 2.142\)
• Bottom left: \(-0.428 + 12 = 11.572\)
• Bottom right: \(-1.714 + 12 = 10.286\)

The result is the matrix:\[\begin{bmatrix} -17.143 & 2.142 \ 11.572 & 10.286 \end{bmatrix}\]Matrix addition is particularly powerful in systems modeling and simulations, where different components often need to be integrated or compared.

Using Graphing Utilities for Matrix Operations

Graphing utilities are incredibly useful tools for visualizing and performing operations with matrices. These digital tools can perform complex calculations quickly and display the results in a clear and visual manner. Essential functions include:

• Executing scalar multiplication and showing step-by-step processes
• Handling matrix addition with ease and speed
• Allowing visualization of matrix transformations

When utilizing a graphing calculator or software:

• You can input the matrix data and apply operations like multiplication and addition seamlessly.
• It's crucial to ensure matrices are entered correctly to match the operations needed.
• The visual output helps in understanding how operations impact the dataset.

Software and utilities such as this turn theoretical calculations into tangible processes, aiding students in conceptualizing abstract algebraic operations. This makes the learning process more interactive and less intimidating for students dealing with matrices.