Page 247 - Probability and Statistical Inference

224 4. Functions of Random Variables and Sampling Distribution

+
of F variable must be ℜ . Note, however, that the random variable U in (4.7.5)
has a continuous distribution.

4.8 Selected Review in Matrices and Vectors

We briefly summarize some useful notions involving matrices and vectors
made up of real numbers alone. Suppose that A m×n is such a matrix having m
rows and n columns. Sometimes, we rewrite A as (a , ..., a ) where a stands
1
i
n
for the i column vector of A, i = 1, ..., n.
th
The transpose of A, denoted by A’, stands for the matrix whose rows
consist of the columns a , ..., a of A. In other words, in the matrix A’, the
1
n
row vectors are respectively a , ..., a . When m = n, we say that A is a square
n
1
matrix.
Each entry in a matrix is assumed to be a real number.
The determinant of a square matrix A n×n is denoted by det(A). If we have
two square matrices A n×n and B n×n , then one has

The rank of a matrix A m×n = (a , ..., a ), denoted by R(A), stands for the
n
1
maximum number of linearly independent vectors among a , ..., a . It can be
1
n
shown that
A square matrix A n×n = (a , ..., a ) is called non-singular or full rank if
1
n
and only if R(A) = n, that is all its column vectors a , ..., a are linearly
1
n
independent. In other words, A n×n is non-singular or full rank if and only if the
n
column vectors a , ..., a form a minimal generator of ℜ .
n
1
A square matrix B n×n is called an inverse of A n×n if and only if AB = BA =
I n×n , the identity matrix. The inverse matrix of A n×n , if it exists, is unique and
–1
it is customarily denoted by A . It may be worthwhile to recall the following
result.
For a matrix A : A exists ⇔ R(A) = n ⇔ det(A) ≠ 0.
–1
n×n
For a 2 × 2 matrix , one has det(A) = a a – a a .
11 22 12 21
Let us suppose that a a ≠ a a , that is det(A) ≠ 0. In this situation, the
11 22
12 21
inverse matrix can be easily found. One has: