Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Use the matrix capabilities of a graphing utility to evaluate the expression. Round your results to the nearest thousandths, if necessary. $$-5\left[\begin{array}{rr} 3.211 & 6.829 \\ -1.004 & 4.914 \\ 0.055 & -3.889 \end{array}\right]-\frac{1}{4}\left[\begin{array}{rr} 1.630 & -3.090 \\ 5.256 & 8.335 \\ -9.768 & 4.251 \end{array}\right]$$

Step-by-Step Solution

Answer

Upon executing the operations accordingly, the solution matrix is \[ \left[\begin{array}{rr} -16.4625 & -33.3725 \ 3.706 & -26.65375 \ 2.167 & 18.38225 \end{array}\right] \]

Step 1: Scale the matrices

Scale the first matrix by -5 and scale the second matrix by -1/4. This results in two new matrices: \[ -5\left[\begin{array}{rr} 3.211 & 6.829 \ -1.004 & 4.914 \ 0.055 & -3.889 \end{array}\right] = \left[\begin{array}{rr} -16.055 & -34.145 \ 5.020 & -24.570 \ -0.275 & 19.445 \end{array}\right] \] and \[ \frac{1}{4}\left[\begin{array}{rr} 1.630 & -3.090 \ 5.256 & 8.335 \ -9.768 & 4.251 \end{array}\right] = \left[\begin{array}{rr} 0.4075 & -0.7725 \ 1.314 & 2.08375 \ -2.442 & 1.06275 \end{array}\right] \]

Step 2: Subtract the Matrices

Subtract the second resultant matrix from the first one, corresponding element by element. This results in a new matrix: \[ \left[\begin{array}{rr} -16.055 & -34.145 \ 5.020 & -24.570 \ -0.275 & 19.445 \end{array}\right] - \left[\begin{array}{rr} 0.4075 & -0.7725 \ 1.314 & 2.08375 \ -2.442 & 1.06275 \end{array}\right] = \left[\begin{array}{rr} -16.4625 & -33.3725 \ 3.706 & -26.65375 \ 2.167 & 18.38225 \end{array}\right] \]

Key Concepts

Scalar Multiplication

Scalar multiplication in the context of matrices involves multiplying every element of a matrix by a constant value, known as a scalar. Imagine you have a matrix and you want to increase or decrease its values consistently. That's where scalar multiplication comes in handy.
For our exercise, consider the first matrix:

• The scalar is -5, which means every element of this matrix is multiplied by -5.
• Similarly, the second matrix has a scalar of \(-\frac{1}{4}\).

Every single value in the matrices is affected by this multiplication, effectively scaling the entire matrix while retaining its original shape. This operation is simple yet powerful, as it allows us to adjust the size of the matrix's elements with precision.

Matrix Subtraction

Matrix subtraction is the process of subtracting each corresponding element of one matrix from another. To perform matrix subtraction, the matrices involved must have the same dimensions, meaning the same number of rows and columns.
In the example, we first scaled both matrices with different scalars, resulting in two new matrices. Then, we subtracted the matrix obtained from scaling the second matrix from the one obtained from scaling the first matrix. This is done element by element:

• Top-left element of the first matrix minus the top-left element of the second matrix,
• top-right of the first minus top-right of the second, and so on.

This is how we get our final matrix after all computations are done.

Graphing Utility

A graphing utility typically refers to a calculator or software that can perform advanced computations, including operations on matrices. Utilizing a graphing utility can significantly simplify the process of evaluating complex matrix expressions.
Here's why graphing utilities are useful:

• They can quickly compute large matrices and handle scalar multiplications and subtractions with ease.
• These tools often come with the capability to visualize the matrices, which is especially helpful in understanding transformations.

When working with matrix operations as in the given exercise, plugging the matrices into a graphing utility ensures precise calculations and provides instant results, saving you time and minimizing errors.

Matrix Evaluation

Matrix evaluation involves computing the final result of matrix operations, such as scalar multiplication and subtraction, as seen in the exercise. It requires understanding the individual matrix operations involved and executing them in the correct order.
The importance of matrix evaluation lies in its ability to quickly provide a comprehensive result from several mathematical operations applied to matrices. Our exercise demonstrates how scaling and subtracting matrices leads to a new matrix that has been thoroughly processed. It's like following a recipe where each step affects the outcome of the dish.
Thus, mastering matrix evaluation is key for solving matrix-related problems, helping you solve complex and otherwise time-consuming calculations efficiently.