# More Exercises on Matrix Operations and Applications

1. Using Cramer’ s rule , find x_{3} in the solution
to the following system of equations:

2. Multiply out the quadratic form in
variables x _{1} and x_{2} defined by

.

3. For the matrix

(a) Compute all the principal minors of A. What do these tell you about whether
A

is positive or negative definite or semidefinite?

(b) Find the eigenvalues by finding all roots of the
characteristic equation |A−rI| =

0. What do these tell you about whether A is positive or negative definite or

semidefinite?

(c) For the principal (largest) eigenvalue r_{1}
of A, find the eigenvector by finding a

solution to the system of equations (A − r_{1}I)h = 0. (Normalize this
eigenvector

to have length 1.)

4. Consider a Markov process with two states in which:

• From state 1, an individual moves to state 2 with
probability 3/4 and remains in

state 1 with probability 1/4;

• From state 2, an individual moves to state 1 with
probability 1/2 and remains in

state 2 with probability 1/2.

For this process, do exercises (a) through (c) below:

(a) What is the transition matrix of this process?.

(b) In a large population whose individuals obey these
transition probabilities in

periods t = 1, 2, . . ., suppose that at t = 1 half the population are in state
1 and

half are in state 2. What is the distribution across states at t = 2?

(c) Use eigenvectors to find a stable state of this Markov
process.

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