S631_Quiz1_Solution

S631_Quiz1_Solution - STAT431/631 Quiz 1(Solution 1 Let A=...

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STAT431/631: Quiz # 1 (Solution) 1. Let A = 4 1 1 2 , B = 2 1 3 1 2 4 (a). Find both AB and B T A T . (5 points) AB = 9 6 16 4 5 11 , B T A T = 9 4 6 5 16 11 (b). Find A - 1 . (5 points) A - 1 = 1 7 2 - 1 - 1 4 (c). For all vectors y y = y 1 y 2 show that y T A y 0. (5 points) y T A y = 4 y 2 1 + 2 y 1 y 2 + 2 y 2 2 = 3 y 2 1 + ( y 1 + y 2 ) 2 + y 2 2 0 2. Let y be an n × 1 column vector y = y 1 y 2 . . . y n (a.) Calculate both y T y and yy T . (5 points) y T y = n X i =1 y 2 i , yy T = y 2 1 y 1 y 2 . . . y 1 y n y 2 y 1 y 2 2 . . . y 2 y n . . . . . . . . . . . . y n y 1 y n y 2 . . . y 2 n (b.) Show that yy T is symmetric. (5 points) ( yy T ) T = ( y T ) T ( y ) T = yy T (c.) Calculate tr ( yy T ). (5 points) tr ( yy T ) = n X i =1 y 2 i 3. Show that I - 1 n J is idempotent. (5 points) ( I - 1 n J )( I - 1 n J ) = I · I - I 1 n J - 1 n J · I + 1 n J 1 n J = I - 1 n J - 1 n J + 1 n J = I - 1 n J therefore,
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