Problem 6 - 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 square matrix. What is its transpose?

Step-by-Step Solution

Answer

The square matrix among the given matrices is Matrix A with dimension \(4×4\). Its transpose, \(A^T\), is: \(\left[\begin{array}{rrrr}2 & -11 & 6 & 5 \\\ -3 & 2 & 0 & 1 \\\ 9 & 6 & 2 & 5 \\\ -4 & 7 & 9 & -8\end{array}\right]\).

Step 1: Identify the square matrix

To find the square matrix, we need to check the dimensions of matrices A, B, C, and D: Matrix A : \(4×4\) Matrix B : \(4×3\) Matrix C : \(1×5\) Matrix D : \(4×1\) A square matrix has the same number of rows and columns. The square matrix among the given matrices is Matrix A with dimension \(4×4\).

Step 2: Compute the transpose

Now, to compute the transpose of the square Matrix A, exchange the rows and columns: \(A^T=\left[\begin{array}{rrrr}2 & -11 & 6 & 5 \\\ -3 & 2 & 0 & 1 \\\ 9 & 6 & 2 & 5 \\\ -4 & 7 & 9 & -8\end{array}\right]\)

Key Concepts

Square Matrix

A square matrix is a special type of matrix in linear algebra where the number of rows (m) is equal to the number of columns (n). This can be represented mathematically as an m × n matrix, with m = n. For instance, if a matrix has three rows and three columns, it would be a 3 × 3 square matrix.
Square matrices play an essential role in various mathematical operations and theories including determinants, eigenvalues, and linear transformations. They can be identified easily due to their symmetrical shape in terms of rows and columns.
Square matrices are crucial in solving systems of linear equations, allowing operations like finding the inverse of a matrix (when it exists). In the provided exercise, Matrix A is a perfect example of a square matrix with dimensions 4 × 4.

Matrix Dimensions

Matrix dimensions are fundamental in understanding matrices. They tell us the size and structure of a matrix and are always written in the form m × n, where m represents the number of rows and n represents the number of columns.

• When a matrix has more columns than rows, it is referred to as a wide matrix.


• If it has more rows than columns, it is known as a tall matrix.


• If the number of rows is equal to the number of columns, as in a square matrix, it has the dimension m = n.

This understanding of matrix dimensions is critical when performing operations such as addition, subtraction, and multiplication, as the dimensions must be compatible for these operations to be well-defined. When examining the matrices in our exercise, we identified Matrix A as having dimensions 4 × 4, confirming it as a square matrix.

Linear Algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. It has a wide range of practical applications in fields such as engineering, physics, computer science, and economics.
Within linear algebra, matrices are used extensively for various computations and transformations. Operations involving matrices include addition, multiplication, finding the determinant, computing inverses, and determining ranks, among others. The concept of transpose, as demonstrated with the square matrix in the exercise, is a fundamental operation where rows and columns are swapped.

• In this exercise, understanding how to transpose Matrix A required basic linear algebra knowledge.


• The transpose of a matrix is important in many applications, including solving linear equations and transforming coordinate systems in graphics programming.

Through mastering linear algebra concepts, students can gain powerful tools for mathematical modeling and problem-solving in diverse scientific and technological domains.