Yes, a unit matrix (also known as an identity matrix) is an orthogonal matrix. Let's delve into why this is the case.

### Definition of Orthogonal Matrix:
A square matrix \( A \) is called orthogonal if its transpose \( A^T \) is equal to its inverse \( A^{-1} \). In mathematical terms, this means:
\[
A^T A = A A^T = I
\]
where \( I \) is the identity matrix.

### Definition of Unit Matrix (Identity Matrix):
A unit matrix (denoted as \( I_n \) for an \( n \times n \) matrix) is a square matrix with ones on the diagonal and zeros elsewhere. For example, a 3x3 identity matrix looks like this:
\[
I_3 = \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{bmatrix}
\]

### Why a Unit Matrix is Orthogonal:
1. **Transpose Equals Inverse:** For the identity matrix \( I_n \), we have:
   \[
   I_n^T = I_n \quad \text{and} \quad I_n^{-1} = I_n
   \]
   Since the transpose of the identity matrix is itself and its inverse is also itself, the identity matrix satisfies the condition \( I_n^T = I_n^{-1} \).

2. **Verification of Orthogonality Condition:** Multiplying the identity matrix by its transpose results in:
   \[
   I_n^T I_n = I_n I_n = I_n
   \]
   This confirms that \( I_n \) is orthogonal.

### Conclusion:
The unit matrix (identity matrix) is orthogonal because it satisfies the condition that its transpose is equal to its inverse. 

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the difference between a unit matrix and a diagonal matrix?
2. Can a non-square matrix be orthogonal?
3. What are some properties of orthogonal matrices?
4. How do you find the inverse of an orthogonal matrix?
5. What is the significance of orthogonal matrices in linear algebra and data analysis?

**Tip:** Remember, the product of two orthogonal matrices is also an orthogonal matrix.

Explore whether a unit matrix (identity matrix) is an orthogonal matrix. Understand the definition of orthogonal matrices, the condition that defines them, and why a unit matrix satisfies this condition. Learn more with detailed explanations and related questions.

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