How many different diagonal matrices of order n can be formed which are involuntary ?

Explanation

To find how many different diagonal matrices of order \( n \) can be formed which are involutory, we will follow these steps:

### Step 1: Understanding Diagonal Matrices
A diagonal matrix is a square matrix where all the elements outside the main diagonal are zero. For a diagonal matrix of order \( n \), it can be represented as:
\[ A = \begin{pmatrix} a_1 & 0 & 0 & \cdots & 0 \\ 0 & a_2 & 0 & \cdots & 0 \\ 0 & 0 & a_3 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & a_n \end{pmatrix} \]
where \( a_1, a_2, \ldots, a_n \) are the diagonal elements.

### Step 2: Involutory Matrices
A matrix \( A \) is said to be involutory if \( A^2 = I \), where \( I \) is the identity matrix. For diagonal matrices, squaring the matrix results in squaring each of the diagonal elements:
\[ A^2 = \begin{pmatrix} a_1^2 & 0 & 0 & \cdots & 0 \\ 0 & a_2^2 & 0 & \cdots & 0 \\ 0 & 0 & a_3^2 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & a_n^2 \end{pmatrix} \]

### Step 3: Setting Up the Equation
Since \( A^2 = I \), we have:
\[ A^2 = \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{pmatrix} \]
This implies:
\[ a_1^2 = 1, \quad a_2^2 = 1, \quad \ldots, \quad a_n^2 = 1 \]

### Step 4: Finding Possible Values for Diagonal Elements
The equation \( a_i^2 = 1 \) has two solutions for each \( a_i \):
\[ a_i = 1 \quad \text{or} \quad a_i = -1 \]
for \( i = 1, 2, \ldots, n \).

### Step 5: Counting the Combinations
Since each diagonal element \( a_i \) can independently take on one of two values (1 or -1), the total number of different diagonal matrices of order \( n \) that can be formed is:
\[ 2 \times 2 \times \ldots \times 2 \quad (n \text{ times}) = 2^n \]

### Conclusion
Thus, the number of different involutory diagonal matrices of order \( n \) is:
\[ \boxed{2^n} \]