Mike University Algebra Worksheet

11
2
Def
Read the following definition carefully, and then answer the questions below.
We define an n × n matrix A to be passive if A2 = A.
Note: before diving in to solving the questions below, you may wish to try to create some
examples of passive matrices (say 2 × 2 to keep it simple).
2.1
Suppose that A is passive. Show that I − A is also passive. Show that A − I is not
passive unless A = I.
2.2
True or False:1 if A is m × n and B is n × m and AB = Im , then BA is passive.
2.3
Show that if A is a passive 2 × 2 matrix with non-zero entries, then the sum of the
entries on the diagonal of A is equal to 1. Create an example of such a matrix and
show that it is indeed
”
# passive.
a b
(That is, if A =
is passive and has a, b, c, d ̸= 0, then a + d = 1.)
c d
3
Def
Read the following definition carefully, and then answer the question below.
We define an n × n matrix B to be meek if B2 = I.
1
Remember, if the statement is True, explain why. If it’s False explain why not, or give an example demonstrating why it’s false (with a
brief explanation). A correct choice of “True” or “False” with no explanation will not receive any credit.
1
Note: as before, start by trying to create some examples of small meek matrices.
True or False: If B is an n × n meek matrix, then the homogeneous system Bx = 0
has only the trivial solution.
2