Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations. \(3\left[\begin{array}{rrr}1 & 1 & -3 \\ 3 & 2 & 3 \\ 7 & -1 & 6\end{array}\right]+4\left[\begin{array}{rrr}-2 & -1 & 8 \\ 4 & 2 & 2 \\ 3 & 6 & 3\end{array}\right]\)

Step-by-Step Solution

Answer

The short answer is: \(\begin{bmatrix} -5 & -1 & 23 \\ 25 & 14 & 17\\ 33 & 21 & 30 \end{bmatrix}\)

Step 1: Multiply the Matrices by the Scalar Values

First, we will multiply each given matrix by its corresponding scalar value. For the first matrix, multiply all elements by 3, and for the second matrix, multiply all elements by 4. 3 * \(\begin{bmatrix} 1 & 1 & -3 \\ 3 & 2 & 3 \\ 7 & -1 & 6 \end{bmatrix} = \begin{bmatrix} 3 & 3 & -9 \\ 9 & 6 & 9\\ 21 & -3 & 18 \end{bmatrix}\) 4 * \(\begin{bmatrix} -2 & -1 & 8 \\ 4 & 2 & 2 \\ 3 & 6 & 3 \end{bmatrix} = \begin{bmatrix} -8 & -4 & 32 \\ 16 & 8 & 8 \\ 12 & 24 & 12 \end{bmatrix}\) Now, we have the two new matrices: \(\begin{bmatrix} 3 & 3 & -9 \\ 9 & 6 & 9\\ 21 & -3 & 18 \end{bmatrix} + \begin{bmatrix} -8 & -4 & 32 \\ 16 & 8 & 8 \\ 12 & 24 & 12 \end{bmatrix}\)

Step 2: Add the Two Resulting Matrices

Now, we need to add the two new matrices together. To do this, simply add the corresponding elements in each matrix: \(\begin{bmatrix} 3 + (-8) & 3 + (-4) & -9 + 32 \\ 9 + 16 & 6 + 8 & 9 + 8\\ 21 + 12 & -3 + 24 & 18 + 12 \end{bmatrix}\) After performing the element-wise addition, the matrix is: \(\begin{bmatrix} -5 & -1 & 23 \\ 25 & 14 & 17\\ 33 & 21 & 30 \end{bmatrix}\) So, the final resulting matrix is: \(\begin{bmatrix} -5 & -1 & 23 \\ 25 & 14 & 17\\ 33 & 21 & 30 \end{bmatrix}\)

Key Concepts

Scalar Multiplication of Matrices

Scalar multiplication is a fundamental matrix operation used commonly in applied mathematics. This operation involves multiplying a constant value, known as a scalar, to each entry within a matrix. Let's imagine we have a scalar, let's say 3, and a matrix \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]. The scalar multiplication of this matrix would be \[3A = \begin{bmatrix} 3a & 3b \ 3c & 3d \end{bmatrix}\]. Every element in the original matrix is multiplied by 3.

In our exercise, each matrix was first multiplied by its respective scalar value. This changed the original matrices to a scaled version of themselves, setting the stage for the next operation, matrix addition.

Matrix Addition

Matrix addition is another vital operation in the matrix calculations toolbox. To add two matrices together, they must have the same dimensions. We do this by adding corresponding elements from each matrix. If \[A = \begin{bmatrix} a1 & b1 \ c1 & d1 \end{bmatrix}\] and \[B = \begin{bmatrix} a2 & b2 \ c2 & d2 \end{bmatrix}\], their sum C = A + B is \[C = \begin{bmatrix} a1 + a2 & b1 + b2 \ c1 + c2 & d1 + d2 \end{bmatrix}\].

In our textbook example, after multiplying the matrices by the scalars, we performed matrix addition by adding the corresponding elements of the scaled matrices to reach our final solution.

Element-wise Matrix Operations

Element-wise operations, such as addition or multiplication, involve performing operations between corresponding elements of two matrices of the same size. This is different from matrix multiplication where we perform a series of multiplications and additions.

For instance, consider matrices \[A = \begin{bmatrix} a1 & b1 \ c1 & d1 \end{bmatrix}\] and \[B = \begin{bmatrix} a2 & b2 \ c2 & d2 \end{bmatrix}\]. The element-wise addition of A and B would be \[A \oplus B = \begin{bmatrix} a1 + a2 & b1 + b2 \ c1 + c2 & d1 + d2 \end{bmatrix}\]. In the exercise, we performed element-wise addition after scaling the matrices.

Applied Mathematics

Matrix operations are not just academic; they are a cornerstone of applied mathematics, which extends into various fields including engineering, physics, computer science, and economics. The techniques used to manipulate matrices, such as the operations discussed—scalar multiplication, matrix addition, and element-wise operations—are used in solving linear equations, transforming geometric data, and even in algorithms that power our internet searches and social media feeds.

In the exercise provided, we executed operations that are applied in real-world scenarios, such as transforming coordinates in computer graphics or adjusting financial models in econometrics. Understanding these operations deepens one's ability to engage with complex systems and improve problem-solving skills in a practical context.