Problem 28 - Advanced Engineering Mathematics

Question

Write the given sum as a single-column matrix. $$ \left(\begin{array}{rrr} 1 & -3 & 4 \\ 2 & 5 & -1 \\ 0 & -4 & -2 \end{array}\right)\left(\begin{array}{r} 3 \\ 2 \\ -1 \end{array}\right)+\left(\begin{array}{r} -1 \\ 1 \\ 4 \end{array}\right)-\left(\begin{array}{r} 2 \\ 8 \\ -6 \end{array}\right) $$

Step-by-Step Solution

Answer

Final matrix: \( \begin{pmatrix} -10 \\ 10 \\ 4 \end{pmatrix} \).

Step 1: Matrix Multiplication

Multiply the given 3x3 matrix by the 3x1 column vector. This involves taking each row of the matrix and performing the dot product with the column vector.- First row: \( 1 \times 3 + (-3) \times 2 + 4 \times (-1) = 3 - 6 - 4 = -7 \)- Second row: \( 2 \times 3 + 5 \times 2 + (-1) \times (-1) = 6 + 10 + 1 = 17 \)- Third row: \( 0 \times 3 + (-4) \times 2 + (-2) \times (-1) = 0 - 8 + 2 = -6 \)The resulting column matrix from this multiplication is:\[ \begin{pmatrix} -7 \ 17 \ -6 \end{pmatrix} \]

Step 2: Adding First Vector

Add the resulting column matrix from Step 1 to the first 3x1 column vector given in the problem.\[ \begin{pmatrix} -7 \ 17 \ -6 \end{pmatrix} + \begin{pmatrix} -1 \ 1 \ 4 \end{pmatrix} = \begin{pmatrix} -7 + (-1) \ 17 + 1 \ -6 + 4 \end{pmatrix} = \begin{pmatrix} -8 \ 18 \ -2 \end{pmatrix} \]

Step 3: Subtract Second Vector

Subtract the second 3x1 column vector from the result obtained in Step 2.\[ \begin{pmatrix} -8 \ 18 \ -2 \end{pmatrix} - \begin{pmatrix} 2 \ 8 \ -6 \end{pmatrix} = \begin{pmatrix} -8 - 2 \ 18 - 8 \ -2 - (-6) \end{pmatrix} = \begin{pmatrix} -10 \ 10 \ 4 \end{pmatrix} \]

Step 4: Final Result

Summarize the computation and provide the final single-column matrix result. The simplified form of the sum as a single-column matrix is:\[ \begin{pmatrix} -10 \ 10 \ 4 \end{pmatrix} \]

Key Concepts

Matrix Multiplication

Matrix multiplication is a way to combine two matrices to produce another matrix. This process is common in linear algebra computations. To multiply a matrix by a column vector, you follow a specific procedure:

• Identify the rows in the matrix and the single column in the vector.
• Perform a dot product between each row of the matrix and the column vector.
• The result for each row gives you an entry in the resulting column matrix.

For example, take a 3x3 matrix and a 3x1 column vector. When these are multiplied, the result is a 3x1 matrix. For each entry in the resulting column matrix, calculate the sum of products of each element in the matrix's row with the corresponding element in the column vector.
This method allows us to transform data in various applications such as graphics and system transformations.

Column Vector

A column vector is a matrix that consists of a single column of elements. It plays a crucial role in linear algebra, simplifying complex operations such as transformations and rotations.

• Column vectors are often used to represent physical quantities like velocities or forces.
• Each element in a column vector can be seen as a coordinate or a dimension magnitude.

In matrix operations, column vectors are frequently used in multiplication. Because of their straightforward mechanism of representing data, they're handy in various algorithmic calculations and simulations. When added or subtracted from other column matrices, the operations are applied element-wise, maintaining clarity and consistency.

Matrix Addition

Matrix addition involves adding two matrices of the same dimensions element by element, resulting in a new matrix of the same size. It's a simple yet powerful operation that helps combine information.

• Ensure that both matrices have identical dimensions. This is imperative for carrying out matrix addition.
• Add each corresponding element from the two matrices to create the elements of the new matrix.

In this process, each result element is simply the sum of the corresponding input elements. Matrix addition is commutative, meaning the order of addition does not matter, and associative, so the grouping of matrices does not affect the sum. This operation is essential in network theory, optics, and other engineering and scientific fields.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings between these spaces. It deals with concepts that are the backbone of many mathematical applications in real life and technology.

• Involves operations on matrices and vectors, forming the core tools for solving linear equations.
• Central in fields like computer science, physics, and engineering, especially in graphics, machine learning, and data analysis.

By employing techniques from linear algebra, complex systems can be simplified into more manageable problems. It provides methods for understanding and solving problems through concepts like transformation matrices and eigenvalues, giving insight into the behavior of linear systems.