Problem 10 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Refer to the following matrices: \(A=\left[\begin{array}{rr}-1 & 2 \\ 3 & -2 \\ 4 & 0\end{array}\right] \quad B=\left[\begin{array}{rr}2 & 4 \\ 3 & 1 \\ -2 & 2\end{array}\right]\) \(C=\left[\begin{array}{rrr}3 & -1 & 0 \\ 2 & -2 & 3 \\ 4 & 6 & 2\end{array}\right] \quad D=\left[\begin{array}{rrr}2 & -2 & 4 \\ 3 & 6 & 2 \\\ -2 & 3 & 1\end{array}\right]\) Compute \(2 A-3 B\).

Step-by-Step Solution

Answer

The short answer is: \(2 A-3 B =\left[\begin{array}{rr}-8 & -8 \\\ -3 & -7 \\\ 14 & -6\end{array}\right]\)

Step 1: Multiply matrix A by scalar 2

We start by multiplying each element of matrix A by the scalar 2: \( A = \left[\begin{array}{rr}-1 & 2 \\\ 3 & -2 \\\ 4 & 0\end{array}\right] \) \(2A = \left[\begin{array}{rr}2(-1) & 2(2) \\\ 2(3) & 2(-2) \\\ 2(4) & 2(0)\end{array}\right] = \left[\begin{array}{rr}-2 & 4 \\\ 6 & -4 \\\ 8 & 0\end{array}\right]\)

Step 2: Multiply matrix B by scalar -3

Next, we multiply each element of matrix B by the scalar -3: \( B =\left[\begin{array}{rr}2 & 4 \\\ 3 & 1 \\\ -2 & 2\end{array}\right] \) \(-3B = \left[\begin{array}{rr}-3(2) & -3(4) \\\ -3(3) & -3(1) \\\ -3(-2) & -3(2)\end{array}\right] = \left[\begin{array}{rr}-6 & -12 \\\ -9 & -3 \\\ 6 & -6\end{array}\right]\)

Step 3: Add the resulting matrices of 2A and -3B

Finally, we add the resulting matrices of 2A and -3B: \(2A - 3B = \begin{bmatrix}-2 & 4 \\\ 6 & -4 \\\ 8 & 0\end{bmatrix} + \begin{bmatrix}-6 & -12 \\\ -9 & -3 \\\ 6 & -6\end{bmatrix}\) \(2A - 3B = \begin{bmatrix}(-2)+(-6) & 4+(-12) \\\ 6+(-9) & (-4)+(-3) \\\ 8+6 & 0+(-6)\end{bmatrix} = \begin{bmatrix}-8 & -8 \\\ -3 & -7 \\\ 14 & -6\end{bmatrix}\) So the final answer is: \(2 A-3 B =\left[\begin{array}{rr}-8 & -8 \\\ -3 & -7 \\\ 14 & -6\end{array}\right]\)

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in linear algebra that involves multiplying every element of a matrix by a single number, known as a scalar. Understanding scalar multiplication is crucial for manipulating matrices, as it represents how matrices can be scaled up or down.

Here's how it works:

• Take each element of the matrix and multiply it by the scalar.
• Ensure that every element from the original matrix is involved in the operation.
• Result in a new matrix of the same dimensions, where each element is the product of the original element and the scalar.

For example, given a matrix \[ A = \begin{bmatrix} -1 & 2 \ 3 & -2 \ 4 & 0 \end{bmatrix} \]and a scalar - let's say 2 - you would multiply each element by 2, providing the resulting matrix \[ 2A = \begin{bmatrix} 2(-1) & 2(2) \ 2(3) & 2(-2) \ 2(4) & 2(0) \end{bmatrix} = \begin{bmatrix} -2 & 4 \ 6 & -4 \ 8 & 0 \end{bmatrix} \].

This operation is similar to distributing the scalar across all elements of the matrix, maintaining the same configuration of the array while changing its size or scale.

Matrix Addition

Matrix addition is a process where we add two matrices by adding their respective elements. To add matrices, they must be of the same dimensions, meaning they have the same number of rows and columns.

Steps for matrix addition:

• Ensure both matrices are of the same dimensions.
• Add the elements in the corresponding positions in each matrix.
• Write the sum in the same position in a new matrix.

For example, if you add matrices \[ A = \begin{bmatrix} -2 & 4 \ 6 & -4 \ 8 & 0 \end{bmatrix} \] and \[ B = \begin{bmatrix} -6 & -12 \ -9 & -3 \ 6 & -6 \end{bmatrix} \], the resulting matrix is \[ A + B = \begin{bmatrix} (-2)+(-6) & 4+(-12) \ 6+(-9) & (-4)+(-3) \ 8+6 & 0+(-6) \end{bmatrix} = \begin{bmatrix} -8 & -8 \ -3 & -7 \ 14 & -6 \end{bmatrix} \].

Matrix addition is similar to adding two sets of numbers: the resulting matrix retains the size and shape of the initial matrices but with newly computed values.

Linear Algebra

Linear algebra is an area of mathematics that studies vectors, matrices, and linear transformations. It forms a crucial foundation for numerous applications in science and engineering, Big Data, machine learning, and network theory.

Key elements of linear algebra include:

• Vectors and Matrices: These are the building blocks, where vectors represent points in space and matrices describe linear transformations.
• Matrix Operations: Including addition, subtraction, and multiplication (such as scalar multiplication), which help transform and manipulate data.
• Determinants and Eigenvalues: Used to understand matrices further, identifying properties like invertibility and stability.

Linear algebra is incredibly vital because it helps solve systems of linear equations, optimize computations, and understand the behavior of complex networks. Its principles are applied in various fields, making it a pivotal part of modern computational techniques. As you delve into matrix operations like scalar multiplication and matrix addition, you're exploring the influential world of linear algebra, impacting everything from graphics to data algorithms.