Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

The transpose of matrix \(\left[\begin{array}{rrr}1 & -1 & 2 \\\ 3 & 4 & 2 \\\ 0 & 1 & 0\end{array}\right]\) is \(\left[\begin{array}{rrr}1 & 3 & 0 \\\ -1 & 4 & 1 \\\ 2 & 2 & 0\end{array}\right]\).

Step 1: Write down the given matrix

Write down the given matrix, which is: \(\left[\begin{array}{rrr}1 & -1 & 2 \\\ 3 & 4 & 2 \\\ 0 & 1 & 0\end{array}\right]\).

Step 2: Transpose the matrix

Interchange its rows and columns to form the transpose of the matrix. This can be done as follows: 1. Row 1 of the original matrix becomes column 1 of the transposed matrix. 2. Row 2 of the original matrix becomes column 2 of the transposed matrix. 3. Row 3 of the original matrix becomes column 3 of the transposed matrix.

Step 3: Write down the transposed matrix

The transpose of the original matrix would be: \(\left[\begin{array}{rrr}1 & 3 & 0 \\\ -1 & 4 & 1 \\\ 2 & 2 & 0\end{array}\right]\).

Key Concepts

Linear Algebra

Linear algebra serves as a foundational stone in the world of mathematics and underpins various applications in science and engineering. At its core, linear algebra deals with vectors, vector spaces, and linear transformations. It's particularly concerned with solving systems of linear equations, which are equations that involve some product of variables and constants, but without exponents. The interactions between variables can be represented by matrices, making matrix operations a crucial aspect of linear algebra.

Understanding the concept of a matrix transpose is an important part of linear algebra. It's akin to flipping axes in a coordinate system. Just as you might flip a book to view it from another angle, transposing a matrix gives you a different perspective on the relationship between the rows and columns. While the exercise given involves a practical application of finding the transpose, grasping the underlying theory of linear algebra could greatly enhance students' comprehension of why this operation is essential and how it can be applied to solve more complex problems.

Matrix Operations

Transposition as a Fundamental Operation

Matrix operations are the bread and butter of manipulating matrix algebra. They include addition, subtraction, multiplication, and transposition, to name a few. Transposing a matrix is one such elementary operation that doesn't just rearrange numbers—it transforms how we interpret data in the context of linear transformations. Let's take a closer look at the exercise provided:

When we are asked to find the transpose of a matrix, we're carrying out an operation that converts all the rows into columns and vice versa. By doing this, the first row \(1, -1, 2\) becomes the first column, the second row \(3, 4, 2\) becomes the second column, and so forth. This operation preserves the diagonal of the matrix and rotates the off-diagonal elements around it. Understanding this procedure is paramount because the transpose of a matrix is widely used in areas such as computer graphics, statistics, and physics.

Transformation Matrices

Transposes and Their Role in Transformations

In the realm of linear algebra, transformation matrices are powerful tools because they allow us to apply linear transformations to objects. This means you can scale, rotate, reflect, or shear objects in a consistent way. The matrix transpose plays a subtle, yet significant role in these transformations.

Matrices are often used as a representation of transformations, and the transpose of a matrix can sometimes reveal intriguing properties of this transformation, such as symmetry. For example, when dealing with orthogonal matrices, which represent rotations and reflections without rescaling, their transpose is their inverse. This implies that reversing an orthogonal transformation is as simple as transposing its matrix. Knowing how to correctly transpose a matrix, as shown in the exercise, is therefore not just a matter of algorithmic procedure, but of understanding the deeper implications it has on geometry and object manipulation within space.