Problem 26 - 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. $$\frac{4}{5}\left[\begin{array}{rr} 14 & -11 \\ -22 & 19 \end{array}\right]+7\left[\begin{array}{rr} -22 & 20 \\ 13 & 6 \end{array}\right]$$

Step-by-Step Solution

Answer

The result of the given matrix operation is \[\left[\begin{array}{cc} -142.800 & 131.200 \ 73.400 & 57.200\end{array}\right]\]

Step 1: Scalar Multiplication

Firstly, the scalar multiplication is done for each matrix. It's done by multiplying each element in the matrix by its scalar. For the first matrix, each element is multiplied by \(\frac{4}{5}\), and for the second matrix, each value is multiplied by 7:First Matrix:\[\frac{4}{5} *\left[\begin{array}{rr} 14 & -11 \ -22 & 19 \end{array}\right] = \left[\begin{array}{cc} 11.2 & -8.8 \ -17.6 & 15.2\end{array}\right]\]Second Matrix:\[7 * \left[\begin{array}{rr} -22 & 20 \ 13 & 6 \end{array}\right] = \left[\begin{array}{cc} -154 & 140 \ 91 & 42\end{array}\right]\]

Step 2: Addition of Matrices

In this step, add the two matrices obtained from scalar multiplication:\[\left[\begin{array}{cc} 11.2 & -8.8 \ -17.6 & 15.2\end{array}\right] + \left[\begin{array}{cc} -154 & 140 \ 91 & 42\end{array}\right] = \left[\begin{array}{cc} -142.8 & 131.2 \ 73.4 & 57.2\end{array}\right]\]

Step 3: Rounding Results

Since the final results are required to the nearest thousandths, each number is rounded, leading to our final matrix:\[-142.8 \rightarrow -142.800\]\[131.2 \rightarrow 131.200\]\[73.4 \rightarrow 73.400\]\[57.2 \rightarrow 57.200\]

Key Concepts

Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a scalar value, which is a constant number. Let's take an example where we need to perform scalar multiplication with the matrix:
\[\frac{4}{5}\begin{bmatrix} 14 & -11 \ -22 & 19 \end{bmatrix}\]
The scalar, in this case, is \( \frac{4}{5} \). Each element of the matrix will be individually multiplied by \( \frac{4}{5} \).

• The first element: \(14\) becomes \(14 \times \frac{4}{5} = 11.2\)
• Continuing similarly for each element in the matrix, you will transform the entire matrix.

This step is necessary to adjust the magnitude of the entire matrix by a uniform factor, which prepares it for further operations like addition.

Matrix Addition

Matrix addition is a straightforward process, but it requires the matrices to be of the same size. In matrix addition, corresponding elements of each matrix are added together.
Consider two matrices from the example:
\[\begin{bmatrix} 11.2 & -8.8 \ -17.6 & 15.2 \end{bmatrix}\]
\[\begin{bmatrix} -154 & 140 \ 91 & 42 \end{bmatrix}\]
You add the top-left element of the first matrix to the top-left element of the second matrix, and continue this for each position:

• \(11.2 + (-154) = -142.8\)
• \(-8.8 + 140 = 131.2\)
• \(-17.6 + 91 = 73.4\)
• \(15.2 + 42 = 57.2\)

The resulting matrix will have the same dimensions as the original matrices, and matrix addition allows for the combination of different datasets succinctly.

Graphing Utility

A graphing utility is a powerful tool for solving matrix operations efficiently and accurately. With a graphing calculator or software, you can easily input matrices and perform operations like scalar multiplication and addition.

• Input the matrices as they appear in the problem.
• Use the software's functions to multiply and add the matrices.
• Obtain results instantly, even for complex calculations.

Using a graphing utility saves time and minimizes errors, making it a staple tool for students tackling matrix operations. It also has graphical capabilities, which help in visualizing the impact of transformations like scaling and addition on a matrix.

Rounding

Rounding is the process of eliminating less significant digits, making the number easier to work with. Often in mathematics, particularly when precise numbers become cumbersome, rounding is essential.
In matrix problems, after performing operations, results might have decimal places. For example, the number \(131.2\) rounded to the nearest thousandth becomes \(131.200\).

• Identify the last necessary digit according to required precision.
• Adjust this digit up or down based on the following number.

Rounding plays a critical role in ensuring that final answers meet presentation standards, especially in printed outputs or when using the results in further calculations.

Algebra

Algebraic principles underpin matrix operations, allowing us to manipulate, analyze, and solve equations.
Matrices can be seen as algebraic objects, where operations follow specific rules analogous to arithmetic operations.

• Scalar multiplication is equivalent to distributing a constant factor over an expression.
• Matrix addition is similar to adding like terms in an equation.

By understanding algebraic concepts, the logic behind matrix operations becomes clearer, and you can see how these operations help solve systems of equations and model real-world scenarios.