Problem 37 - Discrete Mathematics with Applications

Question

The transpose of a matrix \(A=\left(a_{i j}\right)_{m \times n},\) denoted by \(A^{\mathrm{T}},\) is defined as \(A^{\mathrm{T}}=\left(a_{j i}\right)_{n \times m} \cdot\) Find the transpose of each. $$\left[\begin{array}{lll} a & b & c \\ d & e & f \\ f & g & h \end{array}\right]$$

Step-by-Step Solution

Answer

The transpose of the given matrix A is: $$A^T=\left[\begin{array}{lll} a & d & f \\\ b & e & g \\\ c & f & h \end{array}\right]$$

Step 1: Identify the given matrix A

The given matrix A is a 3x3 matrix represented as: $$\left[\begin{array}{lll} a & b & c \\\ d & e & f \\\ f & g & h \end{array}\right]$$

Step 2: Swap rows and columns to form the transpose matrix

To get the transpose of this matrix, swap rows and columns as described earlier: the element at row 1 and column 1 in the original matrix will stay in the same position; the element at row 1 and column 2 in the original matrix will be at row 2 and column 1 in the transpose, and so on. The transpose matrix, denoted by A^T, is represented as: $$A^T=\left[\begin{array}{lll} a & d & f \\\ b & e & g \\\ c & f & h \end{array}\right]$$

Step 3: Write down the final answer

The transpose of the given matrix A is: $$A^T=\left[\begin{array}{lll} a & d & f \\\ b & e & g \\\ c & f & h \end{array}\right]$$

Key Concepts

Matrix Operations

Matrix operations are fundamental techniques in linear algebra that allow us to manipulate matrices to extract useful information or solve complex problems. One of the basic matrix operations is the transpose of a matrix. The transpose is a specific operation where we interchange the rows and columns of a given matrix. This creates a new matrix with its rows representing the original columns and its columns representing the original rows.

• To transpose a matrix, take every element from row positions and swap it with its corresponding column position.
• The mathematical notation for the transpose is usually denoted as \( A^T \) for a given matrix \( A \).
• This operation is particularly useful in various applications like simplifying expressions, solving equations, and more in both mathematics and applied fields.

For a matrix \( A \) that is represented as a 3x3 matrix:
\[ A = \begin{bmatrix} a & b & c \ d & e & f \ f & g & h \end{bmatrix} \]
The transposed matrix \( A^T \) will look like this:
\[ A^T = \begin{bmatrix} a & d & f \ b & e & g \ c & f & h \end{bmatrix} \]
It's a straightforward yet powerful tool to transform matrices for further operations.

Linear Algebra

Linear algebra is an essential branch of mathematics focusing on vector spaces, linear mappings between such spaces, and the representation of these mappings as matrices. Transposing matrices is one of the key operations encountered in linear algebra.

• Linear algebra deals with vectors as rows or columns in matrices, making the transposition operation central to understanding matrix-based calculations.
• The properties of matrix transposition help to simplify complex linear transformations, making them easier to analyze and solve.
• For instance, a symmetric matrix is one that is equal to its transpose: \( A = A^T \). This property is often seen in geometry and computer graphics.

In linear algebra,matrices represent systems of linear equations, transformations, and even data in various fields such as economics, engineering, and physics — highlighting their significance in understanding broader mathematical concepts.

Matrix Representation

Matrix representation is crucial in displaying and working with data in a structured format. The arrangement of elements within a matrix determines how various operations, like the transpose, are performed.

• A matrix is usually defined by its rows and columns—as in an \( m \times n \) matrix format.
• Each element can be addressed by its specific row and column indices, noted as \( a_{ij} \) for the element in the i-th row and j-th column.
• Ensuring correct representation is vital for operations like the transpose, which requires precise row-to-column conversion.

With the exercise given, the matrix:
\[ \begin{bmatrix} a & b & c \ d & e & f \ f & g & h \end{bmatrix} \]
shows how individual elements are moved to different positions to form its transpose. Keeping structural clarity helps maintain data integrity and facilitates further operations like addition, subtraction, or multiplications.