Problem 1 - Algebra and Geometry: Introduction to Linear Algebra for Science & Engineering

Question

Determine which of the following matrices are symmetric. (a) \(A=\left[\begin{array}{rr}0 & 2 \\ 2 & -1\end{array}\right]\) (b) \(B=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]\) (c) \(C=\left[\begin{array}{rrr}1 & 2 & 1 \\ -2 & 1 & 2 \\ -1 & -2 & 1\end{array}\right]\) (d) \(D=\left[\begin{array}{rrr}0 & -1 & 1 \\ -1 & 0 & -1 \\ 1 & -1 & 0\end{array}\right]\)

Step-by-Step Solution

Answer

A, B, and D are symmetric matrices.

Step 1: Definition of a Symmetric Matrix

A matrix is symmetric if it is equal to its transpose. Specifically, a matrix \( M \) is symmetric if \( M = M^T \). This means that the element in the \(i ext{th}\) row and \(j ext{th}\) column is equal to the element in the \(j ext{th}\) row and \(i ext{th}\) column, for all \(i\) and \(j\).

Step 2: Check if Matrix A is Symmetric

Matrix \(A\) is given as \(\begin{bmatrix} 0 & 2 \ 2 & -1 \end{bmatrix}\). The transpose of \(A\), \(A^T\), is \(\begin{bmatrix} 0 & 2 \ 2 & -1 \end{bmatrix}\). Since \(A = A^T\), matrix \(A\) is symmetric.

Step 3: Check if Matrix B is Symmetric

Matrix \(B\) is \(\begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}\). Its transpose \(B^T\) is also \(\begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix}\). Since \(B = B^T\), matrix \(B\) is symmetric.

Step 4: Check if Matrix C is Symmetric

Matrix \(C\) is \(\begin{bmatrix} 1 & 2 & 1 \ -2 & 1 & 2 \ -1 & -2 & 1 \end{bmatrix}\). The transpose is \(\begin{bmatrix} 1 & -2 & -1 \ 2 & 1 & -2 \ 1 & 2 & 1 \end{bmatrix}\). Since \(C eq C^T\), matrix \(C\) is not symmetric.

Step 5: Check if Matrix D is Symmetric

Matrix \(D\) is \(\begin{bmatrix} 0 & -1 & 1 \ -1 & 0 & -1 \ 1 & -1 & 0 \end{bmatrix}\). The transpose is \(\begin{bmatrix} 0 & -1 & 1 \ -1 & 0 & -1 \ 1 & -1 & 0 \end{bmatrix}\). Since \(D = D^T\), matrix \(D\) is symmetric.

Key Concepts

Matrix Transpose

The transpose of a matrix is a fundamental concept in linear algebra. It involves flipping a matrix over its diagonal, which means turning every row of the matrix into a column.This operation is usually denoted by the symbol \(^T\) following the matrix. For example, if we have a matrix \( M = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), its transpose \( M^T \) would be \( \begin{bmatrix} a & c \ b & d \end{bmatrix} \).
The transpose operation does not change the diagonal elements of a square matrix, but it switches the non-diagonal ones.
The transpose is especially important in determining whether a matrix is symmetric, which occurs when the matrix equals its own transpose, staying unchanged.

Matrix Equality

Matrix equality is a simple yet crucial concept in linear algebra. Two matrices are said to be equal if they have the same dimensions and their corresponding elements match perfectly.
Let's say we have two matrices \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \). These matrices are equal because both have 2 rows and 2 columns, and each respective element is identical.
For a matrix to be symmetric, it needs to be equal to its transpose. This equality ensures that the matrix mirrors itself along the diagonal, making it symmetric.

• Dimension similarity: Both matrices must have the same number of rows and columns.
• Element equality: Each pair of corresponding elements must be identical.

Linear Algebra Basics

Linear algebra involves studying vectors, matrices, and the linear mappings between them. It is a foundational branch of mathematics with applications across various fields such as engineering, computer science, economics, and more.
Understanding matrices is one of the basic skills in this field. They can be used to represent systems of equations, transformations, and much more in a concise way. Among many types of matrices, symmetric matrices hold a special place.
A symmetric matrix is useful in many computational systems, especially those involving physical simulations and optimizations.

• Vectors: Arrays of numbers that represent points in space.
• Matrices: Rectangular arrays of numbers expressed in rows and columns.
• Linear mappings: Functions that preserve vector space operations.

For students just getting started with linear algebra, grasping the concept of matrix operations such as addition, multiplication, and, importantly, transpose, becomes vital. This foundation enables exploring more complex topics like eigenvalues and eigenvectors later on.