Problem 11 - Discrete Mathematics with Applications

Question

Let \(A=\left[\begin{array}{ccc}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{ccc}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right]\) . Find each. $$2 A+3 B$$

Step-by-Step Solution

Answer

The short answer is: \(2A + 3B = \left[\begin{array}{ccc}{2} & {-6} & {13} \\\ {0} & {4} & {9}\end{array}\right]\).

Step 1: Perform Scalar Multiplication on A and B

Multiply each element in matrix A by 2 and each element in matrix B by 3: $$ 2A = 2\cdot\left[\begin{array}{ccc}{1} & {0} & {-1} \\\ {0} & {2} & {3}\end{array}\right] = \left[\begin{array}{ccc}{2} & {0} & {-2} \\\ {0} & {4} & {6}\end{array}\right] $$ $$ 3B = 3\cdot\left[\begin{array}{ccc}{0} & {-2} & {5} \\\ {0} & {0} & {1}\end{array}\right] = \left[\begin{array}{ccc}{0} & {-6} & {15} \\\ {0} & {0} & {3}\end{array}\right] $$

Step 2: Add the Resulting Matrices Together

Add the matrices obtained in Step 1 element-wise. That is, add the corresponding elements of each matrix: $$ 2A + 3B = \left[\begin{array}{ccc}{2} & {0} & {-2} \\\ {0} & {4} & {6}\end{array}\right] + \left[\begin{array}{ccc}{0} & {-6} & {15} \\\ {0} & {0} & {3}\end{array}\right] $$ $$ 2A + 3B = \left[\begin{array}{ccc}{2+0} & {0+(-6)} & {(-2)+15} \\\ {0+0} & {4+0} & {6+3}\end{array}\right] $$ $$ 2A + 3B = \left[\begin{array}{ccc}{2} & {-6} & {13} \\\ {0} & {4} & {9}\end{array}\right] $$ Hence, the result of \(2A + 3B\) is the matrix \(\left[\begin{array}{ccc}{2} & {-6} & {13} \\\ {0} & {4} & {9}\end{array}\right]\).

Key Concepts

Scalar Multiplication

Scalar multiplication in matrix operations involves multiplying each element within a matrix by the same fixed number, known as the scalar. This operation is fundamental in linear algebra and is vital for scaling matrices in various applications. For example, if you have a matrix A and you want to double its size, you would multiply each of its elements by 2.
Let's delve into an example to better understand this concept. Given a matrix:

• A = \(\begin{bmatrix}1 & 0 & -1 \0 & 2 & 3\end{bmatrix}\)

To perform scalar multiplication with a scalar of 2, multiply each element in the matrix A by 2, resulting in:

• \(2A = \begin{bmatrix} 2 \cdot 1 & 2 \cdot 0 & 2 \cdot (-1) \ 2 \cdot 0 & 2 \cdot 2 & 2 \cdot 3 \end{bmatrix} = \begin{bmatrix} 2 & 0 & -2 \ 0 & 4 & 6 \end{bmatrix}\)

Scalar multiplication is simple yet powerful, as it allows matrices to be easily scaled up or down by multiplying with positive or negative scalars, respectively.

Matrix Addition

Matrix addition involves adding two matrices by adding their corresponding elements. The matrices involved in addition must have the same dimensions — that is, they must have the same number of rows and columns.
For instance, if we have two matrices A and B, where:

• A = \(\begin{bmatrix}2 & 0 & -2 \0 & 4 & 6\end{bmatrix}\)
• B = \(\begin{bmatrix}0 & -6 & 15 \0 & 0 & 3\end{bmatrix}\)

To perform matrix addition, simply add the elements at matching positions in both matrices. For example:

• \(\begin{bmatrix} 2+0 & 0+(-6) & -2+15 \ 0+0 & 4+0 & 6+3 \end{bmatrix} = \begin{bmatrix} 2 & -6 & 13 \ 0 & 4 & 9 \end{bmatrix}\)

Matrix addition is straightforward and is often used to combine results from different matrix operations, such as when scaling matrices and then summing them together.

Linear Algebra

Linear algebra is the study of vectors, vector spaces, and linear transformations. It provides the foundation for performing various matrix operations, including scalar multiplication and matrix addition. In essence, linear algebra helps in solving systems of linear equations and transforming geometric shapes, making it indispensable in fields like engineering, physics, computer science, and economics.
Consider scalar multiplication, which is a way to scale vectors or matrices within a vector space. Or, think about matrix addition, which combines matrices that live in the same vector space. Both of these operations are guided by the principles of linear algebra.
Key applications of linear algebra include:

• Solving Linear Equations: Essential in mathematical modeling and simulations.
• Transformations: Understanding rotations, scaling, and translations in spaces like 2D and 3D graphics.
• Data Analysis: Techniques such as Principal Component Analysis (PCA) that reduce dimensional data for simplification and insight.

Linear algebra efficiently handles matrices and vectors, allowing problem-solving across various domains and ensuring precision in mathematical computations.