Q1. Verify whether the following matrix A is orthogonal

A= 1/2[2212 - 12- 122].

To determine whether the given matrix \( A \) is orthogonal, we need to verify that the matrix satisfies the orthogonality condition:

\[ A \cdot A^T = I \]

Where:
- \( A^T \) is the transpose of matrix \( A \).
- \( I \) is the identity matrix of the same dimension as \( A \).

Given:
\[ A = \frac{1}{2} \begin{bmatrix} 2 & 2 & 1 \\ 2 & -1 & 2 \\ -1 & 2 & 2 \end{bmatrix} \]

Let's compute \( A \cdot A^T \) and check if it equals the identity matrix.

### Step 1: Find \( A^T \) (the transpose of \( A \))

The transpose of \( A \) is obtained by swapping the rows and columns:

\[ A^T = \frac{1}{2} \begin{bmatrix} 2 & 2 & -1 \\ 2 & -1 & 2 \\ 1 & 2 & 2 \end{bmatrix} \]

### Step 2: Compute \( A \cdot A^T \)

\[
A \cdot A^T = \frac{1}{2} \begin{bmatrix} 2 & 2 & 1 \\ 2 & -1 & 2 \\ -1 & 2 & 2 \end{bmatrix} \cdot \frac{1}{2} \begin{bmatrix} 2 & 2 & -1 \\ 2 & -1 & 2 \\ 1 & 2 & 2 \end{bmatrix}
\]

\[
= \frac{1}{4} \begin{bmatrix} (2 \cdot 2 + 2 \cdot 2 + 1 \cdot 1) & (2 \cdot 2 + 2 \cdot -1 + 1 \cdot 2) & (2 \cdot -1 + 2 \cdot 2 + 1 \cdot 2) \\ (2 \cdot 2 + -1 \cdot 2 + 2 \cdot 1) & (2 \cdot 2 + -1 \cdot -1 + 2 \cdot 2) & (2 \cdot -1 + -1 \cdot 2 + 2 \cdot 2) \\ (-1 \cdot 2 + 2 \cdot 2 + 2 \cdot 1) & (-1 \cdot 2 + 2 \cdot -1 + 2 \cdot 2) & (-1 \cdot -1 + 2 \cdot 2 + 2 \cdot 2) \end{bmatrix}
\]

Simplifying each element:

\[
= \frac{1}{4} \begin{bmatrix} (4 + 4 + 1) & (4 - 2 + 2) & (-2 + 4 + 2) \\ (4 - 2 + 2) & (4 + 1 + 4) & (-2 - 2 + 4) \\ (-2 + 4 + 2) & (-2 - 2 + 4) & (1 + 4 + 4) \end{bmatrix}
\]

\[
= \frac{1}{4} \begin{bmatrix} 9 & 4 & 4 \\ 4 & 9 & 0 \\ 4 & 0 & 9 \end{bmatrix}
\]

\[
= \begin{bmatrix} \frac{9}{4} & 1 & 1 \\ 1 & \frac{9}{4} & 0 \\ 1 & 0 & \frac{9}{4} \end{bmatrix}
\]

Since \( A \cdot A^T \) does not equal the identity matrix \( I \), \( A \) is **not** an orthogonal matrix.

### Would you like a more detailed explanation, or do you have any questions?

Here are some related questions you might find useful:
1. How do you compute the inverse of a matrix?
2. What are the properties of orthogonal matrices?
3. How can you verify if a matrix is invertible?
4. What is the significance of the determinant of a matrix?
5. How do eigenvalues relate to orthogonality?

**Tip:** To verify if a matrix is orthogonal, always check if the product of the matrix and its transpose equals the identity matrix.

Learn how to verify whether matrix A = 1/2[2 2 1; 2 -1 2; -1 2 2] is orthogonal using matrix multiplication and the orthogonality condition. Understand the steps involved in computing A * A^T and comparing it with the identity matrix. Suitable for advanced undergraduate students and mathematics enthusiasts.

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