Final answer:
The statement is false. Not every elementary permutation matrix satisfies P2 = 1.
Step-by-step explanation:
The statement is false. Not every elementary permutation matrix satisfies P2 = 1.
An elementary permutation matrix is a square matrix obtained by interchanging two rows (or columns) of the identity matrix. Let's take an example:
Consider the matrix P = | 0 1 |, which is obtained by interchanging the first and second rows of the 2x2 identity matrix.
To find P2, we need to multiply P by itself:
P x P = | 0 1 | x | 0 1 | = | 0 1 | x | 0 1 | = | 1 0 |.
Therefore, P2 = | 1 0 | ≠ 1, so the statement is false.