This practice is meant to get all students familiar with
the very basic ope rations and terminology
which we will use throughout the course. You can do this by hand or using a
computer. Software
for matrix algebra includes R, S-Plus, Matlab, and Mathematica .
1. Matrix operations : For the matrices be low , find A − B,
AC, and B'A.
2. Matrix characteristics: Linear dependence .
(a) Are the columns of A (below) linearly dependent ? Justify your answer.
(b) Is A an invertible matrix? Justify your answer.
(c) Verify that C (below) is the inverse of B (below).
3. Regression models in matrix notation: An observational
study is being d one in youth smok-
ers. 
is each youth’s average number of
cigarettes smoked per day. The first covarate being
considered is the youth’s age in years (so 
is
the age). The second covariate being consid-
ered is whether or not the youth participates in after-school youth groups, such
as athletic
teams or community service groups (so 
is 1
for yes and 0 for no). Consider the following
regression equations for the first 6 youth in the study (i = 1, 2, 3, 4, 5, 6):
Re-write these equations as one matrix equation. Clearly define each matrix in
your equation.
4. Quadratic forms : Suppose we de termine that
participation in youth groups is not a significant
predictor of cigarette smoking, so the covariate
is dropped from the model. Now we just
have one covariate for age. Least squares (or
maximum likelihood ) estimation of a simple
linear regression 
is done by finding those
and 
which
minimize
( equivalently ,
). Re-write this summation in
matrix notation.
