If `I` is a unit matrix of order `10`, then the determinant of `I` is equal to

Options

A. 10
B. 1
C. `(1)/(10)`
D. 9

Answer

To find the determinant of the unit matrix (identity matrix) of order 10, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Identity Matrix**:
The identity matrix of order `n` is a square matrix of size `n x n` where all the diagonal elements are 1, and all off-diagonal elements are 0. For example, the identity matrix of order 10, denoted as `I10`, looks like this:
\[ I_{10} = \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix} \]

2. **Calculating the Determinant**:
The determinant of an identity matrix of any order is always equal to 1. This can be shown by calculating the determinant for smaller identity matrices:
- For `I1` (1x1 matrix):
\[ \text{det}(I_1) = 1 \]
- For `I2` (2x2 matrix):
\[ I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \quad \Rightarrow \quad \text{det}(I_2) = 1 \cdot 1 - 0 \cdot 0 = 1 \]
- For `I3` (3x3 matrix):
\[ I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \quad \Rightarrow \quad \text{det}(I_3) = 1 \cdot (1 \cdot 1 - 0 \cdot 0) = 1 \]
- Continuing this pattern, we find that the determinant of `I_n` is always 1.

3. **Conclusion**:
Therefore, the determinant of the identity matrix of order 10 is:
\[ \text{det}(I_{10}) = 1 \]

### Final Answer:
The determinant of the identity matrix of order 10 is equal to **1**.

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