Problem 34 - Advanced Engineering Mathematics

Question

In Problems 31-34, suppose \(\mathbf{A}=\left(\begin{array}{rr}2 & 4 \\ -3 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}4 & 10 \\ 2 & 5\end{array}\right)\). Verify the given property by computing the left and right members of the given equality. $$ (6 A)^{T}=6 A^{T} $$

Step-by-Step Solution

Answer

The property \((6 A)^{T}=6 A^{T}\) is verified as true.

Step 1: Calculate \(6 \mathbf{A}\)

First, multiply each element of the matrix \(\mathbf{A}\) by 6. \[6 \mathbf{A} = 6 \begin{pmatrix} 2 & 4 \ -3 & 2 \end{pmatrix} = \begin{pmatrix} 12 & 24 \ -18 & 12 \end{pmatrix}\]

Step 2: Compute the transpose of \((6 \mathbf{A})\)

Find the transpose of the matrix obtained in Step 1 by swapping its rows and columns. \[(6 \mathbf{A})^{T} = \begin{pmatrix} 12 & -18 \ 24 & 12 \end{pmatrix}\]

Step 3: Calculate \(\mathbf{A}^{T}\)

Find the transpose of the matrix \(\mathbf{A}\) by swapping its rows and columns. \[\mathbf{A}^{T} = \begin{pmatrix} 2 & -3 \ 4 & 2 \end{pmatrix}\]

Step 4: Multiply \(6\) with \(\mathbf{A}^{T}\)

Multiply each element of the transposed matrix \(\mathbf{A}^{T}\) by 6. \[6 \mathbf{A}^{T} = 6 \begin{pmatrix} 2 & -3 \ 4 & 2 \end{pmatrix} = \begin{pmatrix} 12 & -18 \ 24 & 12 \end{pmatrix}\]

Step 5: Compare \((6 \mathbf{A})^{T}\) with \(6 \mathbf{A}^{T}\)

Check whether the matrices from Step 2 and Step 4 are identical. Since \((6 \mathbf{A})^{T} = \begin{pmatrix} 12 & -18 \ 24 & 12 \end{pmatrix}\) and \(6 \mathbf{A}^{T} = \begin{pmatrix} 12 & -18 \ 24 & 12 \end{pmatrix}\), they are equal.

Key Concepts

matrix multiplication

Matrix multiplication is a key operation in linear algebra and involves combining two matrices to produce a new matrix. Unlike regular multiplication, where order does not matter, matrix multiplication is not commutative. This means that multiplying matrix \(\mathbf{A}\) by matrix \(\mathbf{B}\) is generally not the same as multiplying matrix \(\mathbf{B}\) by matrix \(\mathbf{A}\).
When we speak of matrix multiplication, it's important to remember that not all matrices can be multiplied together. The number of columns in the first matrix must equal the number of rows in the second matrix. For example, if matrix \(\mathbf{A}\) is 2x3 and matrix \(\mathbf{B}\) is 3x2, they can be multiplied because the inner dimensions match.
In matrix multiplication, the element at the \((i, j)\) position of the resulting matrix is calculated by taking the dot product of the \(i\)-th row of the first matrix and the \(j\)-th column of the second matrix. This process involves multiplying corresponding elements and summing those products. For a simple 2x2 example:

• Multiply the rows of the first matrix with the columns of the second.
• Add the products to get the elements of the new matrix.

This unique way of combining matrices leads to rich applications in computer graphics, system modeling, and more.

matrix properties

In linear algebra, understanding the properties of matrices is fundamental. Properties refer to certain consistent behaviors or qualities that matrices exhibit, which are critical in simplifying and solving problems.
One key property is the distributive property of matrix operations, which states that \(\mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{A}\mathbf{B} + \mathbf{A}\mathbf{C}\) and \((\mathbf{B} + \mathbf{C})\mathbf{A} = \mathbf{B}\mathbf{A} + \mathbf{C}\mathbf{A}\).
Another important property is the associative property of multiplication, given by \((\mathbf{A}\mathbf{B})\mathbf{C} = \mathbf{A}(\mathbf{B}\mathbf{C})\). This means when you are performing multiple matrix multiplications, the order of multiplication does not matter.
The transpose property is crucial, especially when dealing with the equality like \((6\mathbf{A})^{T} = 6 \mathbf{A}^{T}\), as shown in the exercise. This illustrates how scalars can be moved outside of a transpose operation, confirming the scalar multiplication rule for transposes, \((c\mathbf{A})^{T} = c\mathbf{A}^{T}\), where \(c\) is a scalar like 6.
These properties simplify computational work by reducing complex operations into simpler, manageable steps. Knowing these properties allows one to solve various linear algebra problems effectively.

linear algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear mappings between these spaces. It is the backbone of many mathematical concepts and applications.
At its core, linear algebra is concerned with the study of lines, planes, and subspaces, but its reach extends far beyond. It provides the language and framework for understanding linear transformations, which are functions that map one vector space to another while preserving the operations of vector addition and scalar multiplication.
Key concepts include:

• Vectors: Objects that have both magnitude and direction.
• Matrix: A rectangular array of numbers or expressions.
• Systems of linear equations: Equations in which variables appear linearly.
• Determinants: A special number calculated from a square matrix.
• Eigenvalues and eigenvectors: Values that indicate how matrices can stretch or compress space.

Linear algebra's power is seen in diverse applications, from computer science algorithms to machine learning and data analysis. It aids in understanding more advanced mathematical disciplines and fuels the technology that powers data operations today.
In simple terms, linear algebra helps us work with and understand multi-dimensional data, making sense of how different dimensions interact and influence one another.