Problem 25 - Advanced Engineering Mathematics

Question

In Problems 25-28, write the given sum as a single-column matrix. $$ 4\left(\begin{array}{r} -1 \\ 2 \end{array}\right)-2\left(\begin{array}{l} 2 \\ 8 \end{array}\right)+3\left(\begin{array}{r} -2 \\ 3 \end{array}\right) $$

Step-by-Step Solution

Answer

\( \begin{array}{r} -14 \\ 1 \end{array} \)

Step 1: Distribute the Scalars

Begin by multiplying each scalar with its respective matrix. For the first matrix, multiply 4 by each entry:\[4 \left( \begin{array}{r} -1 \ 2 \end{array} \right) = \begin{array}{r} 4(-1) \ 4(2) \end{array} = \begin{array}{r} -4 \ 8 \end{array} \]For the second matrix, multiply -2 by each entry:\[-2 \left( \begin{array}{l} 2 \ 8 \end{array} \right) = \begin{array}{r} -2(2) \ -2(8) \end{array} = \begin{array}{r} -4 \ -16 \end{array} \]For the third matrix, multiply 3 by each entry:\[3 \left( \begin{array}{r} -2 \ 3 \end{array} \right) = \begin{array}{r} 3(-2) \ 3(3) \end{array} = \begin{array}{r} -6 \ 9 \end{array} \]

Step 2: Addition of the Matrices

Add the resulting matrices together by summing the corresponding entries:\[\left( \begin{array}{r} -4 \ 8 \end{array} \right) + \left( \begin{array}{r} -4 \ -16 \end{array} \right) + \left( \begin{array}{r} -6 \ 9 \end{array} \right) = \begin{array}{r} -4 + (-4) + (-6) \ 8 + (-16) + 9 \end{array}\]Calculate each entry:\[= \begin{array}{r} -4 - 4 - 6 \ 8 - 16 + 9 \end{array} = \begin{array}{r} -14 \ 1 \end{array} \]

Step 3: Write the Final Column Matrix

The final result is a single column matrix:\[\begin{array}{r} -14 \ 1 \end{array}\] This matrix represents the sum of the original expression after distribution and addition.

Key Concepts

Scalar Multiplication

Scalar multiplication in linear algebra involves multiplying a matrix by a scalar (a single number). Each element of the matrix is scaled by the value of the scalar. In this exercise, we multiply scalars with the matrices:

• The scalar 4 is multiplied with the matrix \(\begin{array}{c} -1 \ 2 \end{array}\) resulting in \(\begin{array}{c} -4 \ 8 \end{array}\).
• The scalar -2 is applied to the matrix \(\begin{array}{c} 2 \ 8 \end{array}\), giving \(\begin{array}{c} -4 \ -16 \end{array}\).
• Lastly, the scalar 3 is distributed to the matrix \(\begin{array}{c} -2 \ 3 \end{array}\), resulting in \(\begin{array}{c} -6 \ 9 \end{array}\).

This operation needs to be performed on each and every entry of the matrices, which is a key principle in linear algebra.

Matrix Addition

Matrix addition requires adding corresponding elements of two or more matrices. For this, all matrices involved must have the same dimensions (same number of rows and columns).In the given exercise, we added matrices after performing scalar multiplication, shown as:

• For the first element: \(-4 + (-4) + (-6) = -14\).
• For the second element: \(8 + (-16) + 9 = 1\).

The result is a new single-column matrix: \(\begin{array}{c} -14 \ 1 \end{array}\).This process highlights the parallel between algebraic addition and matrix operations, which can be helpful for visual learners.

Column Matrix

A column matrix (or column vector) is a matrix with only one column. Despite having just one column, a column matrix can have one or more rows, meaning it stacks numerical values vertically.In the exercise, we worked out all calculations to eventually express the result as a column matrix:

• The final column matrix derived was \(\begin{array}{c} -14 \ 1 \end{array}\).

Column matrices are commonly used in systems of linear equations, vectors, and transformations, playing a key role in various linear algebra applications.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors and matrices, and the operations performed on them, such as addition, multiplication, and transformation. Key components learned through this sort of problem include:

• Understanding how to manipulate matrices with basic arithmetic operations.
• The skill of reading and interpreting matrix notation accurately.
• The application of these skills in computational algorithms and data analysis.

Whether handling equations of planes in space or handling datasets, linear algebra forms the foundation for many fields within computational sciences and engineering.