ST512
1.
Quiz1
SSII-2010
(40 pts.) Below are two matrices A and B. Find, if possible, each requested item. If not
possible, briefly explain why:
 4 1
A   2 1
 1 1
 3 2 1
B

 1 0 1 
a) (8 pts)
A+B 
b) (8 pts)
BA 
c) (8 pts)
A + B 
d) (8 pts)
B1 
e) Rank of the matrix B is
2. (24 pts.) Here are two new matrices A and B.
1
1
A
1

1
0 0 3
1 0 0 
1 1 0

1 1 1
 0.2 1.2 0.6 0.6 
 0.2 0.2 0.6 0.6 

B= 
 0
1
1
0 


 0.4 0.4 0.2

a) (8 pts.) Fill in the missing element in B to make B the inverse of A. (Note, AA1  I )
b) (8 pts.) Compute, if possible, the rank of matrix B.
rank  B   _______
c) (8 pts.) Compute, if possible, from the given information, B1 
If not possible, explain.
July 8, 2010
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ST512
Quiz1
SSII-2010
3. (36 pts.) Let Gi denote the weight gain for the firstborn male piglet in litter i over a given
period of time and let Pi denote the number of other piglets (siblings) in the same litter. I
observed n litters of pigs and fit a least squares line relating first born male weight gain G
to number of siblings P so my model was Gi  0  1Pi  ei . The estimated model came
out to Gˆ  80  0.5P . My n 2 X matrix had a column of 1s and a column of Pi values as
i
i
usual. I obtained
 50 110 
XX  

110 260
Answer these if possible from this information (if not possible, put NP)
(a) (7 pts.) How many observations (litters) did I have in my experiment? n = ____
(b) (7 pts.) Compute, if possible, the sum of the n weight gains in this experiment
n
G
i 1
i

(c) (9 pts.) Compute 90  XX  
1
(d) (13 pts.) We have seen that the estimated intercept b 0 and slope b1 together form a random
vector that varies from sample to sample with variance-covariance matrix  XX   2 . Assuming
1
 2 is known to be 90, compute the covariance between b 0 and b1 , as well as the variance of W ,
where W  2b0  3b1  18 .
July 8, 2010
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