If I₃ is the identity matrix of order 3, then (I₃)⁻¹ is
A. 0
B. 3I₃
C. I₃
D. Not necessarily exists.

We are given the identity matrix \( I_3 \) of order 3 and asked to determine the inverse of \( I_3 \), denoted as \( (I_3)^{-1} \).

The identity matrix \( I_3 \) is a 3x3 matrix, defined as:

\[
I_3 = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\]

By definition, an identity matrix \( I_n \) of order \( n \) has the property that:

\[
I_n \cdot A = A \cdot I_n = A \quad \text{for any matrix} \ A
\]

Also, the inverse of a matrix \( A \), denoted as \( A^{-1} \), satisfies:

\[
A \cdot A^{-1} = A^{-1} \cdot A = I_n
\]

For the identity matrix, we have:

\[
I_3 \cdot I_3 = I_3
\]

This means that the inverse of \( I_3 \) is \( I_3 \) itself.

Thus, the inverse of \( I_3 \) is \( I_3 \), and the correct answer is:

C. \( I_3 \)