Step 1: Understanding the Question:
The question asks for the determinant of an identity matrix, denoted as \(I_n\), for any size \(n \times n\).
Step 2: Key Formula or Approach:
A key property of determinants is that for any upper triangular, lower triangular, or diagonal matrix, the determinant is simply the product of the elements on its main diagonal.
Step 3: Detailed Explanation:
An identity matrix \(I_n\) is a special type of diagonal matrix. Its main diagonal consists entirely of 1s, and all other elements are 0s.
For example, the \(3 \times 3\) identity matrix is:
\[ I_3 =
\begin{bmatrix}
1 & 0 & 0
0 & 1 & 0
0 & 0 & 1
\end{bmatrix}
\]
To find its determinant, we apply the rule for diagonal matrices and multiply the elements on the main diagonal.
\[ \det(I_n) = \underbrace{1 \times 1 \times 1 \times \dots \times 1}_{n \text{ times}} \]
The product of any number of 1s is always 1.
\[ \det(I_n) = 1 \]
Step 4: Final Answer:
The determinant of an identity matrix of any order \(n\) is always 1.