- Covered today: (NOTE: this is not in the book) Matrix algebra.
- For Next time:
- 1.
- In (a)-(c) let
Find each as indicated
- (a)
-
- (b)
-
- (c)
-
, (, a scalar)
- 2.
- Find and for the matrix
- 3.
- Find the general formula for
- (a)
-
- (b)
-
- 4.
- A matrix is called an involution if (I is identity
matrix). Give an example of a
matrix (not equal to I) that is an
involution.
- 5.
- A is called an idempotent if . Give an example of a matrix that is an idempotent ( not equal I or 0).
- 6.
- Prove that
hold for
matrices.
- 7.
- Find the inverse of the matrix
- 8.
- Prove that
hold for matrices.
- 9.
- Find the inverse of the matrix and the condition needed
for the inverse to exist.
- 10.
- Find the value of for which
has no inverse.
- 11.
- Use the method of inverses, if possible, to solve each system:
- (a)
-
- (b)
-
- 12.
- Consider the linear system
, where
and
Determine conditions on the constants
, and so that
- (a)
- the rank of is 2,
- (b)
- the rank of is 1, but the rank of
is 2
- (c)
- the rank of and the rank of
are each 1.
- 13.
- Using the matrix , vector and the results of exercise (12)
to give conditions on
, and so that the linear system has
- (a)
- no solution
- (b)
- one solution
- (c)
- infinitely many solutions.
- 14.
- Use the theorems and definitions learned in class to say as much as
possible concerning the number of solutions of the given linear system without
actually solving it.
- (a)
-
- (b)
-
- (c)
-
- 15.
- Suppose that the augmented matrix of a linear system is given by
For what values of and is there
- (a)
- no solution?
- (b)
- exactly one solution?
- (c)
- infinitely many solutions?
- 16.
- Suppose that the row-reduced echelon form of a linear system
has the following form, where and are arbitrary
constants.
- (a)
- What is the rank of ?
- (b)
- How many solutions does
have?
- (c)
- Is invertible?
- 17.
- Construct the smallest system you can with more unknowns than equations,
but no solution.
- 18.
- Give examples of matrices for which the number of solutions to
is
- (a)
- 0 or 1, depending on b;
- (b)
- infinite, independent of b;
- (c)
- 0 or infinite, depending on b;
- (d)
- 1 regardless of b.
- (e)
- How is the rank of the matrix related to its dimensions in each case?
- 19.
- Let
Determine a set of constrained variables and a set of free variables.
What is the rank of ? Find the general solution to
.
Math 9 Fall 2000
2000-10-25