Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the transpose of each matrix. \(\left[\begin{array}{rrrr}4 & 2 & 0 & -1 \\ 3 & 4 & -1 & 5\end{array}\right]\)

Step-by-Step Solution

Answer

The transpose of the given matrix A is: \[ A^T = \left[\begin{array}{rr} 4 & 3 \\ 2 & 4 \\ 0 & -1 \\ -1 & 5 \end{array}\right] \]

Step 1: Identify the given matrix

The given matrix is: \[ A = \left[\begin{array}{rrrr} 4 & 2 & 0 & -1 \\ 3 & 4 & -1 & 5 \end{array}\right] \]

Step 2: Find the transpose by switching rows and columns

To find the transpose of the matrix A, which we denote as \(A^T\), we need to switch its rows and columns. The first row of A will become the first column of \(A^T\), and the second row of A will become the second column of \(A^T\).

Step 3: Write the transpose matrix

Writing the transpose matrix by switching rows and columns, we get: \[ A^T = \left[\begin{array}{rr} 4 & 3 \\ 2 & 4 \\ 0 & -1 \\ -1 & 5 \end{array}\right] \] The transpose of the given matrix A is: \[ A^T = \left[\begin{array}{rr} 4 & 3 \\ 2 & 4 \\ 0 & -1 \\ -1 & 5 \end{array}\right] \]

Key Concepts

Matrices

Matrix is a fundamental concept in various fields of mathematics and science. In simple terms, a matrix is a rectangular array of numbers arranged in rows and columns. Each element within a matrix is identified by its position in these rows and columns. Matrices are denoted by uppercase letters like \(A\), \(B\), etc. A single number in a matrix is referred to as an element. For example, in our exercise, the given matrix \(A\) consists of 2 rows and 4 columns. This matrix is classified as a 2x4 (read as "2 by 4") matrix because it has 2 rows and 4 columns.
Matrices have various applications including solving systems of linear equations, representing graphs, and transforming geometrical data in graphics. Understanding how matrices function as a data structure is essential in both mathematical theory and practical applications like computer graphics and machine learning.

Matrix Operations

Matrix operations are procedures that can be applied to one or more matrices, resulting in a new matrix or some other output. Among the basic operations are addition, subtraction, multiplication, and the concept of a matrix transpose.
The transpose of a matrix is particularly noteworthy. It involves swapping the rows and columns of the matrix. Mathematically, if matrix \(A\) is comprised of elements \(a_{ij}\) where \(i\) is the row number and \(j\) is the column number, the transpose of \(A\), denoted as \(A^T\), will have elements \(a_{ji}\). This operation effectively "flips" the matrix over its diagonal. In our exercise, the matrix \(A\) is transposed to \(A^T\), where the original rows become columns, resulting in a matrix of size 4x2 from the original 2x4.
Knowing how to transpose a matrix is crucial because it is used in various scenarios such as switching from one coordinate system to another and simplifying complex matrix operations.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vectors, vector spaces, linear transformations, and systems of linear equations. It provides a framework for understanding how various types of mathematical objects interact and transform.
A major component of linear algebra is exploring transformations between vector spaces, often represented succinctly by matrices. For instance, the transpose of a matrix, as discussed in this exercise, plays a role in theoretical and computational aspects of linear algebra. It can, for example, change the representation of the problem, revealing different properties or making certain operations easier to perform. Understanding concepts, such as eigenvalues and eigenvectors, often involves manipulations that include the use of a transpose.
Linear algebra has vast applications in different fields like physics, computer science, and economics. It aids in solving practical problems involving complex systems, optimizing resource allocations, and even in modern innovations such as machine learning algorithms. Thus, mastering linear algebraic techniques is immensely helpful in many scientific and engineering domains.