If a square matrix `A` is involutory, then `A^(2n+1)` is equal to:

Options

A. (a) `I`
B. (b) `A`
C. (c) `A^(2) `
D. (d) `(2n +1) A`

Explanation

To solve the problem, we need to understand the properties of an involutory matrix. An involutory matrix \( A \) satisfies the condition:

\[ A^2 = I \]

where \( I \) is the identity matrix.

Now, we want to find \( A^{2n+1} \).

### Step 1: Express \( A^{2n+1} \)

We can express \( A^{2n+1} \) as follows:

\[ A^{2n+1} = A^{2n} \cdot A \]

### Step 2: Simplify \( A^{2n} \)

Since \( A \) is involutory, we know that:

\[ A^2 = I \]

Thus, we can find \( A^{2n} \) by recognizing that:

\[ A^{2n} = (A^2)^n = I^n = I \]

### Step 3: Substitute \( A^{2n} \) back into the expression

Now we can substitute \( A^{2n} \) back into our expression for \( A^{2n+1} \):

\[ A^{2n+1} = A^{2n} \cdot A = I \cdot A = A \]

### Conclusion

Therefore, we conclude that:

\[ A^{2n+1} = A \]

The correct answer is option B: \( A \).

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