If `I_n` is the identity matrix of order n, then rank of `I_n` is

Options

A. 1
B. n
C. 0
D. none of these

Answer

To find the rank of the identity matrix \( I_n \) of order \( n \), we can follow these steps:

### Step 1: Understand the Identity Matrix
The identity matrix \( I_n \) is a square matrix of order \( n \) with ones on the diagonal and zeros elsewhere. For example:
- For \( n = 2 \):
\[ I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
- For \( n = 3 \):
\[ I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 2: Determine Non-Zero Rows
The rank of a matrix is defined as the maximum number of linearly independent rows (or columns). In the identity matrix \( I_n \):
- Each row contains at least one non-zero element (specifically, a '1' on the diagonal).
- There are \( n \) rows in total.

### Step 3: Count the Non-Zero Rows
Since all \( n \) rows of \( I_n \) contain non-zero elements and are linearly independent, the number of non-zero rows is \( n \).

### Step 4: Conclusion
Thus, the rank of the identity matrix \( I_n \) is equal to \( n \).

### Final Answer
The rank of \( I_n \) is \( n \).

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