The identity matrix \( I \) is a square matrix in which all the diagonal elements are \( 1 \) and all the off-diagonal elements are \( 0 \). For any size \( n \), the \( n \times n \) identity matrix is defined as:

\[
I = \begin{bmatrix} 
1 & 0 & 0 & \cdots & 0 \\ 
0 & 1 & 0 & \cdots & 0 \\ 
0 & 0 & 1 & \cdots & 0 \\ 
\vdots & \vdots & \vdots & \ddots & \vdots \\ 
0 & 0 & 0 & \cdots & 1 
\end{bmatrix}.
\]

### Properties of \( I \):

1. **Idempotence** (\( I^2 = I \)):  
   Multiplying the identity matrix by itself yields the identity matrix:
   \[
   II = I.
   \]
   This holds because the diagonal entries remain \( 1 \) and the off-diagonal entries remain \( 0 \).

2. **Neutral Element**:  
   For any square matrix \( A \) of the same size as \( I \), we have:
   \[
   AI = IA = A.
   \]

### Verification of \( I^2 = I \):

For a \( 2 \times 2 \) identity matrix:
\[
I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.
\]
Multiplying \( I \) by itself:
\[
I^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.
\]

This holds for all square sizes \( n \), confirming that \( I^2 = I \). The property \( II = I \) is why \( I \) is called the **identity** matrix—it acts like multiplying by \( 1 \) in scalar arithmetic.

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Would you like more examples or deeper explanations? Here are 5 related questions for you to explore:

1. What is the role of the identity matrix in linear algebra and matrix multiplication?  
2. Can the identity matrix be used to find the inverse of another matrix?  
3. How does the identity matrix generalize to block matrices or other algebraic systems?  
4. What are the eigenvalues of the identity matrix, and why?  
5. How does the identity matrix differ from the zero matrix?  

**Tip:** The identity matrix is a foundational concept; mastering it helps in solving advanced topics like eigenvalues, matrix diagonalization, and linear transformations.

Learn about the identity matrix, its definition, and key properties like I^2 = I, where multiplying the identity matrix by itself gives the same matrix. This concept is fundamental in linear algebra and matrix operations, ideal for students in Grades 9-12.

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