Q:

# The 3 × 3 matrix P satisfies the matrix equation P^2 = P.(a) What are the possibilities for the determinant of P?(b) Explain why there are no other possibilities.(c) For each possible determinant, give an example of P with that determinant.

Accepted Solution

A:
Answer:  The answers are given below.Step-by-step explanation:  Given that a  3 × 3 matrix P satisfies the matrix equation P² = P.We are to(a) find the possibilities for the determinant of P.(b) explain the reason behind there are no other possibilities.(c) give an example of P, for each possible determinant.(a) According to the given information, we have$$P^2=P\\\\\Rightarrow P^2-P=0\\\\\Rightarrow P(P-I)=0\\\\\Rightarrow P=0,~~~P=I.$$So, P can be either a zero matrix of order 3 or an identity matrix of order 3.If P = 0, then det(P) = 0     and     if P = I, then det(P) = 1.Therefore the possible determinants of P are 0 and 1.(b) There can be any other determinant other than 0 and 1, because if so, then the given equation P² = P will not be satisfied.(c) If |P| = 0, then the matrix P can be can be of the form as follows:$$P=\left[\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right] .$$If |P| = 1, the the matrix P can be of the form as follows :$$P=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] .$$Thus, all the parts are answered.