Problem 25 - 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. $$\frac{3}{7}\left[\begin{array}{rr} 2 & 5 \\ -1 & -4 \end{array}\right]+6\left[\begin{array}{rr} -3 & 0 \\ 2 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The final answer in thousandths is: \(\begin{bmatrix} -17.143 & 2.143 \ 11.571 & 10.286 \end{bmatrix}\).

Step 1: Scalar Multiplication of the First Matrix

Multiply the first matrix \(\begin{bmatrix} 2 & 5 \ -1 & -4 \end{bmatrix}\) by the scalar \(\frac{3}{7}\). Scalar multiplication is done by multiplying each entry of the matrix by the scalar. This gives us: \[\frac{3}{7} \times \begin{bmatrix} 2 & 5 \ -1 & -4 \end{bmatrix} = \begin{bmatrix} \frac{6}{7} & \frac{15}{7} \ -\frac{3}{7} & -\frac{12}{7} \end{bmatrix}.\]

Step 2: Scalar Multiplication of the Second Matrix

Next, multiply the second matrix \(\begin{bmatrix} -3 & 0 \ 2 & 2 \end{bmatrix}\) by the scalar 6. This gives us: \[6 \times \begin{bmatrix} -3 & 0 \ 2 & 2 \end{bmatrix} = \begin{bmatrix} -18 & 0 \ 12 & 12 \end{bmatrix}.\]

Step 3: Addition of the Two Matrices

Now, add the two resulting matrices from steps 1 and 2. This is done by adding corresponding entries in the matrices. This gives us: \[\begin{bmatrix} \frac{6}{7} & \frac{15}{7} \ -\frac{3}{7} & -\frac{12}{7} \end{bmatrix} + \begin{bmatrix} -18 & 0 \ 12 & 12 \end{bmatrix} = \begin{bmatrix} -\frac{120}{7} & \frac{15}{7} \ \frac{81}{7} & \frac{72}{7} \end{bmatrix}.\]

Step 4: Rounding off the results

Finally, round off each element of the result matrix to the nearest thousandths: \[-\frac{120}{7} \approx -17.143, \quad \frac{15}{7} \approx 2.143, \quad \frac{81}{7} \approx 11.571, \quad \frac{72}{7} \approx 10.286.\] So the final answer is: \[\begin{bmatrix} -17.143 & 2.143 \ 11.571 & 10.286 \end{bmatrix}.\]

Key Concepts

Scalar Multiplication

Scalar multiplication in the realm of matrices is a basic yet vital operation carried out between a matrix and a scalar (a single number). It involves multiplying every element of the matrix by the scalar to produce a new matrix.

In the given exercise, scalar multiplication is applied to perform the operation \( \frac{3}{7} \times \begin{bmatrix} 2 & 5 \ -1 & -4 \end{bmatrix} \). The scalar value, \( \frac{3}{7} \), is distributed across each element within the matrix. This action is akin to stretching or shrinking the matrix by the scale of the scalar involved.

For students, it's essential to remember that each element must be multiplied individually -- a straightforward yet sometimes tedious process, particularly with larger matrices or fractions.

Matrix Addition

Moving on to matrix addition, this operation is one in which two matrices of identical dimensions are combined to create a new matrix. In this operation, corresponding elements from each matrix are added together.

The exercise provided illustrates matrix addition through \( \begin{bmatrix} \frac{6}{7} & \frac{15}{7} \ -\frac{3}{7} & -\frac{12}{7} \end{bmatrix} + \begin{bmatrix} -18 & 0 \ 12 & 12 \end{bmatrix} \). Here, similar positions in each matrix are considered for the sum. For example, the top left value of the first matrix is added to the top left value of the second matrix, and so on.

It's important for students to line up the matrices properly to ensure that each element is correctly positioned for the addition. Mixing up these positions can lead to errors in the outcome.

Graphing Utilities

Efficient Matrix Operations with Graphing Utilities

The specific mention of graphing utilities in the exercise recognizes the usefulness of technology in handling complex matrix calculations. These utilities are advanced calculators or software that can perform a multitude of operations including, but not limited to, matrix arithmetic, graph plotting, and statistical analysis.

Students are encouraged to familiarize themselves with graphing utilities as they can greatly expedite the process of matrix operations. These tools are designed to handle the minutiae of calculations, such as scalar multiplication and matrix addition, and often have built-in functions to carry out these tasks with high accuracy and efficiency. This helps to minimize computational errors that might occur with manual calculations.

Rounding Numbers

Rounding numbers, the step that often concludes many mathematical calculations, involves approximating a number to make it shorter or simpler while retaining a value that is close enough to the original for the intended purpose.

In the provided solution, rounding is needed to simplify the results obtained from matrix operations. The exercise stipulates rounding to the nearest thousandths, setting a precision level that strikes a balance between simplicity and accuracy. For students, understanding when and how to round numbers is crucial. It can be the difference between an answer considered 'correct' in a real-world application and one that is overly detailed or too imprecise. Correctly rounding \( -\frac{120}{7} \) to \( -17.143 \) demonstrates this step.

Key to rounding is recognizing the place value to round to and whether to increase the last retained digit based on the next digit (5 or above gets rounded up).