[e] Let Akita, . . . .151,\" he at symmetric idempntent matriees, and that each 3.1- is afsixenxn- Supper-x: A1 +Ag+---+Am = I- Shaw that AiAj = II] whenever i 5'9 j- ThereliIJIreP if z m MVNfI], I], then the cnllecticn 11111. :cTAgz, . . . . :cTAmz are mutuallyr independent. This is l-rnawn as Cerhranis theareIn. Hint: using the spectral decnrnpesitinn them'ern, write AI- = {.1in where L},- is a n. x sl- matrix; here at dennte si = rltIIAI-II- thus L]; is a matrix whdse st calnmns are a suhset cf cnlumns lire-In some nrthngenal matrix. Let s = [it I rs I I as What is LIQT 1? Hanan as 3.. = trial; 1iII'hat then is 2:1 as '3' Is L} a square matrix 'i' What then is LITE} i\" Additinnal Hint: see stactinn 3.1-4 {Cachran's thee-tern} hf A. Agresti \"lendatihns at linear and generalized linear models\" [available nnline thrnugh the .lHIl library}. F'nr the applicatinn at finchran's themem1 see sectihn 3-? df A- Agresti heel-t {in particular sectinn 3-2.2; it is impnrtant tr: nate that the setup in sectian 3.2.2. can easily.r he extended tr.- multiple nesta'l mndels Ms C M1 t: Mg C I: My fer same If, and dennting by P1: the pmjectinn matrices amiated with Indel Mk, it is trivially true that P1+[PaP1}+[P'sPs}+---+lPKPHll+IPH=1- 1What dues Cdchran's thee-rem say ahnnt the test statistic {using sum hf squares] fer mmparing Mk against M; for I: a: E" '1'