MultiStats set 2

# MultiStats set 2 - is also idempotent(b Show that I − X X...

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EDUC771: Applied Multivariate Analysis Problem Set 2 1. The matrices A, B and C are given below: A = 3 5 0 1 2 1 3 7 1 B = 3 1 -1 5 2 0 1 1 -1 (i) Add the two matrices together using SPSS. (ii) Compute the product using SPSS. (iii) Find the determinant of the matrices. Then determine which matrices have an inverse. (iv) For any matrix in part (iii) with an inverse, find it. 2. Determine whether or not the following system of equations has a solution. Explain your answer. 6 x + 8 y = 5 3 x + 4 y = 6 3. Solve the following system of equations using matrix operations. (You may use SPSS). x + y + 2 z = 5 x + y + z = 0 x + 2 y + 4 z = 2 4. (a) If X is a (nxp) idempotent matrix, show that X ( X ' X ) 1 X

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Unformatted text preview: ' is also idempotent. (b) Show that I − X ( X ' X ) − 1 X ' is idempotent as well. (c) Further, if X ' = (1 1 1 1 . .. 1) then show that I − X ( X ' X ) − 1 X ' reduces to I − 1 n 1 1' . 5. If X is any matrix (n x p) using the result that Tr(AB) = Tr(BA), show that Tr [ X ( X ' X ) − 1 X '] = p and deduce that Tr [ I − X ( X ' X ) − 1 X '] = n − p 6. For each of the system of equations below, determine how many solutions that each has, and explain your answer (you may do this using SPSS). (a) 3 x 1 − x 2 = 2 − 6 x 1 + 2 x 2 = − 4 (b) 2 x 1 − 3 x 2 = 5 − 2 x 1 + 3 x 2 = 2...
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MultiStats set 2 - is also idempotent(b Show that I − X X...

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