Problem 5 - Advanced Engineering Mathematics

Question

$$ \text { In Problems } 5-10 \text {, determine whether the given matrix is orthogonal. } $$ $$ \left(\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix is orthogonal because \( A^T A = I \).

Step 1: Understanding Orthogonality

A matrix is orthogonal if its transpose is equal to its inverse. For a matrix \( A \), this is represented by \( A^T = A^{-1} \). Additionally, the product \( A^T A \) should result in the identity matrix \( I \).

Step 2: Transpose the Matrix

Transpose the given matrix. The matrix is: \[A = \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix}\]The transpose of \( A \), denoted as \( A^T \), is obtained by swapping rows with columns. \[A^T = \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix}\]

Step 3: Calculate the Product of Transpose and Original Matrix

Multiply the transpose \( A^T \) with the original matrix \( A \). This product should yield the identity matrix if \( A \) is orthogonal.\[A^T A = \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}\]This result is the identity matrix \( I \).

Step 4: Conclusion on Orthogonality

Since \( A^T A \) results in the identity matrix, the original matrix \( A \) is orthogonal.

Key Concepts

Matrix Transpose

In linear algebra, the transpose of a matrix is a fundamental concept. Transposing a matrix involves flipping the matrix over its diagonal. This effectively means that the row and column indices of each element are switched. For example, if you have a matrix \( A \) with elements \( a_{ij} \), the transpose of \( A \), denoted as \( A^T \), will have elements \( a_{ji} \).
For a given matrix:
\[A = \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix}\]The transpose, \( A^T \), is:
\[A^T = \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix}\]

• Swapping the first row and column, the second row and column, and so forth keeps the diagonal elements the same.
• The process leaves symmetric matrices unchanged because such matrices are identical to their transposes.

Transposing is a simple yet powerful operation that helps in understanding other concepts like orthogonality.

Matrix Inverse

A matrix inverse is a key concept in linear algebra, representing a matrix that reverses another under multiplication, akin to how division works in arithmetic. For a matrix \( A \), an inverse \( A^{-1} \) exists if and only if the product \( A A^{-1} = I \) and \( A^{-1} A = I \), where \( I \) is the identity matrix.
To find an inverse:

• The matrix must be square (same number of rows and columns).
• Its determinant must be non-zero.

The inverse operation is crucial when solving linear systems, as it allows one to solve the equation \( Ax = b \) using \( x = A^{-1}b \). In the case of orthogonal matrices, an exciting feature is that the transpose of an orthogonal matrix is its inverse, simplifying computations immensely.

Identity Matrix

In the world of matrices, the identity matrix is like the number 1 in multiplication. It is a square matrix with ones on the diagonal and zeros elsewhere. For an identity matrix \( I \) of size \( n \times n \), it looks like:
\[I = \begin{pmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{pmatrix}\]Using this special matrix, a fundamental property in matrix multiplication is acknowledged: \( AI = IA = A \). This means multiplying any matrix by an identity matrix leaves it unchanged.

• An identity matrix acts as the neutral element of matrix multiplication.
• It is used in defining determinants, inverse matrix operations, and verifying orthogonality.

When an orthogonal matrix is multiplied by its transpose, the result should be the identity matrix.

Linear Algebra

Linear algebra is an area of mathematics focused on vector spaces and the linear mappings between them. It includes the study of matrices and determinants, systems of linear equations, and vector spaces.
Key concepts include:

• Vectors: Objects representing quantities with both magnitude and direction.
• Matrices: Rectangular arrays that represent linear transformations and can be manipulated to solve equations.
• Systems of equations: Linear algebra provides efficient ways to handle and solve them using matrices.

Matrices like the ones described can represent data, manipulate coordinates, and even change the coordinate system itself through transformations. Understanding linear algebra opens the door to various applications across physics, computer graphics, and economics.