Question: 18.5 Let PX and MX be as defined in Equations (18.24) and (18.25). a. Prove that PXMX = 0n * n and that PX and

18.5 Let PX and MX be as defined in Equations (18.24) and (18.25).

a. Prove that PXMX = 0n * n and that PX and MX are idempotent.

b. Derive Equations (18.27) and (18.28).

c. Show that rank(PX) = k + 1 and rank(MX) = n − k − 1. [Hint:

First solve Exercise 18.10 and then use the fact that trace(AB) =

trace(BA) for conformable matrices A and B.]

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