Question

The 3 × 3 matrix P satisfies the matrix equation P^2 = P.

Answer 1

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.