Explanation:

To understand why the correct answer is option B, let's break down the given options and their definitions:

a) Idempotent: A matrix A is idempotent if A^2 = A, i.e., multiplying the matrix by itself once gives the same matrix.

b) Involutory: A matrix A is involutory if A^2 = I, i.e., multiplying the matrix by itself once gives the identity matrix.

c) Nilpotent of order 2: A matrix A is nilpotent of order 2 if A^2 = 0, i.e., multiplying the matrix by itself once gives the zero matrix.

d) Triangular: A matrix A is triangular if all the entries below or above the main diagonal are zero.

Understanding the Definitions:

In this question, we are given that A^2 = I. This means that when we multiply matrix A by itself, the result is the identity matrix I.

Explanation of Correct Answer:

The correct answer is option B, involutory.

Involutory matrices have the property that when multiplied by themselves, they give the identity matrix. In this case, A^2 = I, which matches the definition of an involutory matrix.

It is important to note that not all idempotent, nilpotent of order 2, or triangular matrices satisfy the given condition A^2 = I.

For example:
- An idempotent matrix satisfies A^2 = A, which is not the same as A^2 = I.
- A nilpotent matrix of order 2 satisfies A^2 = 0, which is also not the same as A^2 = I.
- A triangular matrix does not necessarily satisfy A^2 = I. It only has the property that all the entries below or above the main diagonal are zero.

Hence, the correct answer is option B, involutory, as it is the only option that satisfies the given condition A^2 = I.