Problem 5 - Advanced Engineering Mathematics

Question

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.

Step 1: Understand Orthogonal Matrices

An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors. In simpler terms, 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: Write Down the Given Matrix

We have the matrix:\[A = \begin{pmatrix}0 & 1 & 0 \1 & 0 & 0 \0 & 0 & 1\end{pmatrix}\] We need to find if this is orthogonal by checking if \( A^T A = I \).

Step 3: Calculate the Transpose of the Matrix

To find the transpose \( A^T \), we swap the rows and columns of matrix \( A \):\[A^T = \begin{pmatrix}0 & 1 & 0 \1 & 0 & 0 \0 & 0 & 1\end{pmatrix}\] Note that in this case, \( A^T = A \) because the matrix is symmetric.

Step 4: Compute the Product \( A^T A \)

Calculate the product \( A^T A \):\[A^T A = \begin{pmatrix}0 & 1 & 0 \1 & 0 & 0 \0 & 0 & 1\end{pmatrix}\begin{pmatrix}0 & 1 & 0 \1 & 0 & 0 \0 & 0 & 1\end{pmatrix}\]Perform the matrix multiplication to get:\[A^T A = \begin{pmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{pmatrix} = I\]

Step 5: Conclusion

Since \( A^T A = I \), the matrix \( A \) is orthogonal by definition.

Key Concepts

Matrix Transposition

Matrix transposition is an operation that flips a matrix over its diagonal. Essentially, this means converting its rows into columns. For example, if you have a matrix:\[A = \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix}\]The transpose of matrix \( A \), denoted as \( A^T \), would be:\[A^T = \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix}\]Notice how each row in the original matrix becomes a column in the transposed matrix. In our example, the matrix is symmetric, meaning the transposition does not change the matrix.
This usually happens when a matrix is equal to its own transpose. Transposing a matrix is a key step in numerous mathematical processes, like determining whether a matrix is orthogonal.

Matrix Multiplication

Matrix multiplication involves combining two matrices to produce a third matrix. The key rule to remember is that the number of columns in the first matrix must equal the number of rows in the second matrix. Here's a quick step-by-step to multiply two matrices:

• Multiply each element of the rows of the first matrix with the corresponding elements of the columns of the second matrix.
• Add up the products to get the final value for each element in the resulting matrix.

For instance, to multiply our transposed matrix \( A^T \) with itself in step 4, follow these steps: multiply and add the elements in the corresponding positions. Whenever you perform such calculations correctly, the resulting matrix can sometimes be an identity matrix, which in the case of orthogonal matrices, confirms their orthogonal status.
Matrix multiplication is a powerful operation, but errors can easily occur without careful calculations.

Identity Matrix

An identity matrix, often denoted by \( I \), is a special kind of square matrix. It's defined as having ones along its main diagonal and zeros elsewhere. Here's an example of a 3x3 identity matrix:\[I = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}\]The identity matrix has a special property: when any matrix is multiplied by an identity matrix of compatible dimensions, it remains unchanged. In mathematical terms:

• \( AI = IA = A \)

In the context of orthogonal matrices, like in our exercise, a matrix \( A \) is orthogonal if its transpose multiplied by itself yields the identity matrix (\( A^T A = I \)).
This characteristic serves as proof in confirming that a matrix is indeed orthogonal.