Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Refer to the following matrices: \(A=\left[\begin{array}{rrrr}2 & -3 & 9 & -4 \\ -11 & 2 & 6 & 7 \\ 6 & 0 & 2 & 9 \\ 5 & 1 & 5 & -8\end{array}\right]\) \(B=\left[\begin{array}{rrr}3 & -1 & 2 \\ 0 & 1 & 4 \\ 3 & 2 & 1 \\ -1 & 0 & 8\end{array}\right]\) \(C=\left[\begin{array}{lllll}1 & 0 & 3 & 4 & 5\end{array}\right]\) \(D=\left[\begin{array}{r}1 \\ 3 \\ -2 \\ 0\end{array}\right]\) Identify the column matrix. What is its transpose?

Step-by-Step Solution

Answer

The column matrix is matrix D: \(D= \begin{bmatrix}1 \\ 3 \\ -2 \\ 0 \end{bmatrix}\). To find its transpose, we interchange its rows and columns, which results in \(D^T = \begin{bmatrix}1 & 3 & -2 & 0 \end{bmatrix}\).

Step 1: Examine each given matrix and identify which one has only one column. #Step 2: Find the Transpose of the Column Matrix#

Once the column matrix is identified, interchange the rows and columns to find its transpose.

Step 2: Set up the problem

Write out the given matrices or vectors in standard form.

Step 3: Perform the matrix operations

Execute the required operations, showing key intermediate steps.

Step 4: Analyze the result

Interpret the result in terms of the original problem.

Step 5: State the conclusion

Clearly state the final answer.

Step 6: Conclude with the answer

The column matrix is matrix D: \(D= \begin{bmatrix}1 \\ 3 \\ -2 \\ 0 \end{bmatrix}\). To find its transpose, we interchange its rows and columns, which results in \(D^T = \begin{bmatrix}1 & 3 & -2 & 0 \end{bmatrix}\).

Key Concepts

Matrix Transpose

The transpose of a matrix is an operation that flips a matrix over its diagonal, switching its rows with its columns. This operation is denoted by the superscript 'T' such as in \(A^{T}\), where \(A\) is the original matrix.

For instance, if you have a matrix \(X\), which is a 2x3 matrix given by:
\(X = \left[\begin{array}{ccc}a & b & c \ d & e & f\end{array}\right]\)
its transpose, \(X^{T}\), is a 3x2 matrix and looks like:
\(X^{T} = \left[\begin{array}{cc}a & d \ b & e \ c & f\end{array}\right]\)
This means that the element in the ith row and jth column of \(X\) becomes the element in the jth row and ith column of \(X^{T}\).

Coming back to our exercise, the column matrix \(D\), given by:
\(D = \left[\begin{array}{r}1 \ 3 \ -2 \ 0\end{array}\right]\)
will have its transpose simply by turning its single column into a single row, resulting in:
\(D^{T} = \left[\begin{array}{rrrr}1 & 3 & -2 & 0\end{array}\right]\)
Transposing a matrix is a fundamental operation in linear algebra and is used in various applications including solving systems of linear equations, computing orthogonal projections, and many more.

Matrices Identification

Identifying different types of matrices is a key skill in linear algebra. Matrices come in many forms, including row matrices, column matrices, square matrices, diagonal matrices, etc. Each type has unique properties and applications.

For example:

• A row matrix has only one row.
• A column matrix, which is our focus in this exercise, has only one column.
• A square matrix has an equal number of rows and columns.
• A diagonal matrix is a square matrix where all elements outside the main diagonal are zero.

In the given exercise, we have to identify which one is a column matrix. By definition, a column matrix has exactly one column irrespective of the number of rows. So, looking at our options \(A\), \(B\), \(C\), and \(D\), it's clear that \(D\) is the only column matrix as it has the following form:
\(D = \left[\begin{array}{r}1 \ 3 \ -2 \ 0\end{array}\right]\)
This matrix has multiple rows, but only one column, fitting the definition perfectly. Recognizing matrices properly allows for proper application of various linear algebra techniques including matrix multiplication, finding determinants, and solving linear systems.

Linear Algebra

Linear Algebra is a branch of mathematics concerning linear equations, linear functions, and their representations in vector spaces and through matrices. It is fundamental in advanced mathematics and applications in natural science, engineering, computer science, economics, and more.

For those dealing with linear algebra, it involves various operations on matrices and vectors, which include: addition, subtraction, scalar multiplication, and more complex operations such as finding determinants, inverses, and eigenvalues.

The study of linear algebra involves understanding the structure and properties of vector spaces and the mappings between them, known as linear transformations. Within the context of our exercise, identifying matrix types and performing the transpose are introductory aspects of what linear algebra entails. As students progress, they will delve into more complex concepts such as vector spaces, linear independence, orthogonality, and dimension theory, which all build upon the basic understanding of matrices and their transformations.