SSTT312: Linear Models
Exercise 1
1. Suppose the least square regression line fitted to 100 measurements of head
circumference(𝑦) and gestational age (𝑋) is
𝑦̂ = 3.9143 + 0.7801𝑋
In addition, the following quantities are given.
𝑆𝑒2 = 2.529372 ,
a)
b)
c)
d)
𝑆𝑥𝑥 = 635.2637 ,
𝑆𝛽̂0 = 1.8291
Interpret the slope of the regression line.
Test the null hypothesis that   0 .
Construct a 95% confidence interval for  .
Find the relationship between the correlation coefficient and regression coefficient. Is
testing   0 equivalent to testing   0 ?
6 −2 3
2
5 −1 3
2. Let 𝐀 = (−1
1 0), 𝐂 = ( −1
1 2), 𝐁 = (7
2 −3 5
3
3
2 7
Find the following:
a)
b)
c)
d)
e)
𝐁𝐱
𝐲′𝐁
𝐱 ′ 𝐀𝐱
𝐁′ 𝐁
√𝒚′ 𝒚
3. Use 𝐱, 𝐲, 𝐀 and 𝐁 as defined in the problem above
a) Find 𝐱 + 𝐲 and 𝐱 − 𝐲.
b) Find the trace(𝐀), trace(𝐁) and trace(𝐀 + 𝐁).
c) Find 𝐀𝐁 and 𝐁𝐀.
2
4. Let 𝐀 = (
1
a)
b)
c)
d)
e)
5
),
3
1
𝐁=(
5
−6 2
1
), 𝐈 = (
0 3
0
Show that (𝐀𝐁)′ = 𝐁′ 𝐀′ .
Show that 𝐀𝐈 = 𝐀 and 𝐈𝐁 = 𝐁.
Find the determinant of A.
Find 𝐀−1 .
Find (𝐀−1 )−1and compare with 𝑨.
0
).
1
−3
3
3
4 ), 𝒙 = ( −1 ), 𝒚 = (2).
1
2
4
2
5. Use matric A below to illustrate that trace(𝐀′ 𝐀) = trace(𝐀𝐀′ ) = ∑𝑛𝑖=1 ∑𝑚
𝑗=1 𝑎𝑖𝑗 .
3 1
2
𝐀=(
).
1 0 −1
6. Let 𝐁 =
2
3
0
√2
(3
a)
b)
c)
d)
e)
0
1
√2
3
0
1
3
0 .
)
Find the rank of 𝐁.
Show that 𝐁 is idempotent.
Show that 𝐈 − 𝐁 is idempotent.
Find the trace of 𝐁 i.e. tr(𝐁).
Find the eigenvalues of 𝐀.
1 1 −2
7. Let 𝐂 = ( −1 2
1 ).
0 1 −1
a) Find the eigenvalues of 𝐂. Are they all positive?
b) Find associated normalized eigenvectors.
c) Find tr(𝐂) and det|𝑪|.
d) Verify that trace(𝐂) = ∑3𝑖=1 𝜆𝑖 and 𝑑𝑒𝑡|𝐂| = ∏3𝑖=1 𝜆𝑖 .
8. Given the Matrix
a)
b)
c)
d)
1 2 5 1
𝐁 = (2 2 10 6).
3 4 15 1
Find the number of linearly independent columns of the matrix.
Find the number linearly independent rows of the matrix.
Find the rank of the matrix.
Find the inverse of the following matrix.
3
9. Let 𝐴 = (0
0
0 0
4 0).
0 2
2
a) Show that trace(𝐀′ 𝑨) = ∑𝑛𝑖=1 ∑𝑛𝑗=1 𝑎𝑖𝑗
b) Show that if 𝐀 is idempotent then ( 𝐈 − 𝐀) is idempotent.
c) Let 𝐀 be any n by n matrix and let P be n by n matrix ; then show that
𝑡𝑟𝑎𝑐𝑒(𝐀) = trace(𝐏−1 𝐀𝐏)
10. Let 𝑨 be any n by n matrix with the characteristics roots 𝜆1 , 𝜆2 … 𝜆𝑛 ; then show that
trace(𝑨) = ∑𝑛𝑖=1 𝜆𝑖 .