Problem 7 - Advanced Engineering Mathematics

Question

$$ \text { In Problems } 5-10 \text {, determine whether the given matrix is orthogonal. } $$ $$ \left(\begin{array}{rrr} 0 & 0 & 1 \\ -\frac{12}{13} & \frac{5}{13} & 0 \\ \frac{5}{13} & \frac{12}{13} & 0 \end{array}\right) $$

Step-by-Step Solution

Answer

The given matrix is orthogonal.

Step 1: Understand the Definition of an Orthogonal Matrix

An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors. Mathematically, a matrix \( A \) is orthogonal if \( A^T A = I \), where \( A^T \) is the transpose of \( A \), and \( I \) is the identity matrix.

Step 2: Transpose the Given Matrix

The given matrix is: \[\begin{pmatrix} 0 & 0 & 1 \ -\frac{12}{13} & \frac{5}{13} & 0 \ \frac{5}{13} & \frac{12}{13} & 0 \end{pmatrix} \]To find the transpose, swap rows with columns:\[A^T = \begin{pmatrix} 0 & -\frac{12}{13} & \frac{5}{13} \ 0 & \frac{5}{13} & \frac{12}{13} \ 1 & 0 & 0 \end{pmatrix}\]

Step 3: Multiply the Transpose with the Original Matrix

Calculate \( A^T A \). Multiply the transpose by the original matrix:\[\begin{pmatrix} 0 & -\frac{12}{13} & \frac{5}{13} \ 0 & \frac{5}{13} & \frac{12}{13} \ 1 & 0 & 0 \end{pmatrix}\begin{pmatrix} 0 & 0 & 1 \ -\frac{12}{13} & \frac{5}{13} & 0 \ \frac{5}{13} & \frac{12}{13} & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}\]This product is the identity matrix \( I \).

Step 4: Conclusion

Since \( A^T A \) equals the identity matrix \( I \), the given matrix is orthogonal according to the definition.

Key Concepts

Matrix Transposition

Matrix transposition is a simple yet fundamental operation in linear algebra. It involves flipping a matrix over its diagonal, which means converting the rows of the matrix into columns, and the columns into rows.
This transformation results in a new matrix known as the transpose of the original matrix and is denoted by the superscript "T".

• For example, if you have a matrix \( A \) with elements \( a_{ij} \), its transpose \( A^T \) will have elements \( a_{ji} \).
• The operation does not change the dimensions of the matrix if the original matrix is square.
• Transposition is particularly useful when dealing with orthogonal matrices, where checking whether the transpose multiplied by the original matrix results in an identity matrix is a key step.

The act of transposing can be visualized as simply rotating the elements around the matrix's main diagonal.
This operation is foundational for further matrix operations like multiplication and can be executed effortlessly once understood.

Identity Matrix

An identity matrix is a special kind of square matrix composed of ones on the diagonal and zeros elsewhere. It acts as the neutral element in matrix multiplication, similar to how the number 1 functions in regular multiplication.
The identity matrix is denoted by \( I \), and its size is determined by the number of rows or columns it has.

• For a 3x3 matrix, the identity matrix looks like this: \[\begin{pmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \end{pmatrix}\]
• When a matrix is multiplied by an identity matrix of appropriate size, it remains unchanged—this is akin to multiplying a number by one.
• This unique property is critical in validating whether a given matrix is orthogonal.

In terms of practical use, the identity matrix serves as a benchmark in many computational applications because it maintains the original data of matrices when used correctly.
Thus, understanding how it works is vital for anyone dealing with matrix operations.

Matrix Multiplication

Matrix multiplication is the process of multiplying two matrices to produce a new matrix. It involves combining rows from the first matrix with columns of the second matrix.
This operation has specific rules and conditions:

• The number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be possible.
• Each element in the resulting matrix is computed by taking the dot product of the corresponding row and column vectors.
• The size of the resulting matrix is determined by the number of rows of the first matrix and the number of columns of the second matrix.

Matrix multiplication is associative and distributive but not commutative, which means that the order of multiplication matters.
In the context of orthogonal matrices, multiplying the transpose of a matrix by the original matrix should yield the identity matrix.
This property helps to verify the orthogonality of the matrix and illustrates how matrix multiplication is essential in linear transformations and other mathematical computations.