Problem 14 - Advanced Engineering Mathematics

Question

$$ \begin{aligned} &\text { In Problems } \underline{\phantom{xxx}} , \text { find the entries } c_{23} \text { and } c_{12} \text { for the matrix }\\\ &\mathbf{C}=2 \mathbf{A}-3 \mathbf{B} \end{aligned} $$ $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & -1 & 1 \\ 2 & 2 & 1 \\ 0 & -4 & 1 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{lll} 2 & 0 & 5 \\ 0 & 4 & 0 \\ 3 & 0 & 7 \end{array}\right) $$

Step-by-Step Solution

Answer

\(c_{23} = 2\), \(c_{12} = -2\)

Step 1: Understand the Problem

We need to find the entries \( c_{23} \) and \( c_{12} \) for the matrix \( \mathbf{C} = 2\mathbf{A} - 3\mathbf{B} \). The matrix \( \mathbf{C} \) is obtained by multiplying each element of matrix \( \mathbf{A} \) by 2, each element of matrix \( \mathbf{B} \) by 3, and subtracting the results.

Step 2: Calculate \(2\mathbf{A}\)

Multiply each element of matrix \( \mathbf{A} \) by 2:\[2\mathbf{A} = 2 \times \begin{pmatrix} 1 & -1 & 1 \ 2 & 2 & 1 \ 0 & -4 & 1 \end{pmatrix} = \begin{pmatrix} 2 & -2 & 2 \ 4 & 4 & 2 \ 0 & -8 & 2 \end{pmatrix}\]

Step 3: Calculate \(3\mathbf{B}\)

Multiply each element of matrix \( \mathbf{B} \) by 3:\[3\mathbf{B} = 3 \times \begin{pmatrix} 2 & 0 & 5 \ 0 & 4 & 0 \ 3 & 0 & 7 \end{pmatrix} = \begin{pmatrix} 6 & 0 & 15 \ 0 & 12 & 0 \ 9 & 0 & 21 \end{pmatrix}\]

Step 4: Calculate \(\mathbf{C} = 2\mathbf{A} - 3\mathbf{B}\)

Subtract the matrices from Step 2 and Step 3:\[\mathbf{C} = \begin{pmatrix} 2 & -2 & 2 \ 4 & 4 & 2 \ 0 & -8 & 2 \end{pmatrix} - \begin{pmatrix} 6 & 0 & 15 \ 0 & 12 & 0 \ 9 & 0 & 21 \end{pmatrix} = \begin{pmatrix} 2-6 & -2-0 & 2-15 \ 4-0 & 4-12 & 2-0 \ 0-9 & -8-0 & 2-21 \end{pmatrix}\]This simplifies to:\[\mathbf{C} = \begin{pmatrix} -4 & -2 & -13 \ 4 & -8 & 2 \ -9 & -8 & -19 \end{pmatrix}\]

Step 5: Identify \( c_{23} \) and \( c_{12} \)

In the matrix \( \mathbf{C} \), the entry \( c_{23} \) corresponds to the element in the second row and third column, which is 2.The entry \( c_{12} \) corresponds to the element in the first row and second column, which is -2.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra, where two matrices of the same dimension are combined by adding their corresponding elements. This is an element-wise process, meaning each entry in the resulting matrix is the sum of the entries at the same position in the original matrices.
For example, if we have matrices \( \mathbf{X} \) and \( \mathbf{Y} \) that are both 3x3 matrices, their sum \( \mathbf{Z} = \mathbf{X} + \mathbf{Y} \) is given by adding each corresponding element:

• The element in the first row, first column of \( \mathbf{Z} \) is \( x_{11} + y_{11} \).
• The element in the first row, second column of \( \mathbf{Z} \) is \( x_{12} + y_{12} \).
• This pattern continues throughout the matrix.

A key requirement for matrix addition is that the matrices must have the same dimensions. This ensures that there are like positions across both matrices that can be added together. Also, while the operations involved are simple, mastering the technique is crucial when dealing with matrix equations and transformations.

Matrix Subtraction

Matrix subtraction is very similar to matrix addition, except that instead of summing we subtract the corresponding elements of two matrices. Like addition, subtraction requires the matrices involved to have the same dimensions.
Let's say you have two matrices \( \mathbf{P} \) and \( \mathbf{Q} \), both 3x3. When you find the matrix \( \mathbf{R} = \mathbf{P} - \mathbf{Q} \), each element in \( \mathbf{R} \) is formulated as follows:

• The element in the first row, first column of \( \mathbf{R} \) is \( p_{11} - q_{11} \).
• The element in the first row, second column of \( \mathbf{R} \) is \( p_{12} - q_{12} \).
• This subtraction continues for all corresponding elements in the matrix.

Matrix subtraction is often used when dealing with balance equations, such as when working with net changes or adjustments in systems represented by matrices. Understanding the concept of subtracting matrices positions you well for more complex operations, such as solving systems of linear equations.

Scalar Multiplication

Scalar multiplication in matrix algebra is the process of multiplying every element of a matrix by a single constant (called a scalar). This operation is straightforward and involves distributing the scalar across each element of the matrix.
For instance, consider a matrix \( \mathbf{M} \) and a scalar \( k \). The resulting matrix, \( k \mathbf{M} \), has each element of \( \mathbf{M} \) multiplied by \( k \). More precisely:

• The element in the first row, first column of \( k \mathbf{M} \) is \( k \times m_{11} \).
• The element in the first row, second column of \( k \mathbf{M} \) is \( k \times m_{12} \).
• This process is repeated for every entry in the matrix.

Scalar multiplication can be used as a step in larger computations, such as adjusting the strength or magnitude of some represented quantities in physics or economics. It's an essential tool in linear transformations.

Matrix Elements

Matrix elements are the individual entries found within a matrix. These are organized into rows and columns, and each element is typically denoted by its position.
Consider a simple 3x3 matrix:\(\begin{pmatrix} a & b & c \d & e & f \g & h & i \end{pmatrix}\)

• The element \( a \) is located in the first row, first column, and can be referred to as \( m_{11} \).
• The element \( e \) in the second row, second column can be denoted as \( m_{22} \).
• Each element has a unique position, indicated by its row and column number, such as \( m_{32} \) for the element in the third row, second column.

Understanding matrix elements is crucial for mastering matrix operations, as any addition, subtraction, or substitution you perform with matrices requires precise manipulation of these individual entries. This understanding also leads to more complex topics, such as identifying principal diagonals or using matrices in transformations.