Page 62 - A Course in Linear Algebra with Applications

46 Chapter Two: Systems of Linear Equations

/ l 2 - 3 1 \
(c) 3 1 2 2 .
\ 8 1 9 1 /

2. Put each of the matrices in Exercise 2.2.1 in reduced row
echelon form.
3. Prove that the row operation of type (a) which interchanges
rows i and j can be obtained by a combination of row opera-
tions of the other two types, that is, types (b) and (c).

4. Do Exercises 2.1.1 to 2.1.4 by applying row operations to
the augmented matrices.

5. How many row operations are needed in general to put an
n x n matrix in row echelon form?

6. How many row operations are needed in general to put an
n x n matrix in reduced row echelon form?
7. Give an example to show that a matrix can have more than
one row echelon form.
8. If A is an invertible n x n matrix, prove that the linear
system AX = B has a unique solution. What does this tell
you about the number of pivots of A?

9. Show that each elementary row operation has an inverse
which is also an elementary row operation.