Zero Matrix
How to prove the zero matrix statements: definition, 2 properties, 2 examples, and their solutions.
Definition
Definition
A zero matrix is a matrix
whose elements are all 0.
So AO = OA = O.
Property 1
Property
If AB = O,
then A = O or B = O.
This statement is false.
Unlike the numbers,
AB = O doesn't mean
either A or B is a zero matrix.
Example 1
Example
Solution
Recall that
a counterexample is an example
that makes the statement false.
And the given statement is a conditional statement.
So, set a counterexample
that seems to make
the hypothesis, AB = O, true
and the conclusion, A = O or B = O, false.
Set A = [0 1 / 0 0] and B = [1 0 / 0 0].
First, show that
AB = O is true.
A = [0 1 / 0 0]
B = [1 0 / 0 0]
So AB = [0 1 / 0 0][1 0 / 0 0].
Solve [0 1 / 0 0][1 0 / 0 0].
Multiply Matrices
Row 1, column 1:
0⋅1 + 1⋅0
Row 1, column 2:
0⋅0 + 1⋅0
Row 2, column 1:
0⋅1 + 0⋅0
Row 2, column 2:
0⋅0 + 0⋅0
So
[0 1 / 0 0][1 0 / 0 0]
= [0⋅1 + 1⋅0 0⋅0 + 1⋅0 / 0⋅1 + 0⋅0 0⋅0 + 0⋅0].
0⋅1 + 1⋅0
= 0 + 0
0⋅0 + 1⋅0
= 0 + 0
0⋅1 + 0⋅0
= 0 + 0
0⋅0 + 0⋅0
= 0 + 0
0 + 0 = 0
The elements of the matrix is all 0.
So this matrix is a zero matrix.
You got AB = O.
So the given hypothesis, AB = O,
is true.
Next, show that
A = O or B = O
is false.
A = [0 1 / 0 0]
So A = O is false.
B = [1 0 / 0 0]
So B = O is false.
So
A = O or B = O
is false.
Disjunction
See the given conditional statement.
The hypothesis, AB = O, is true.
The conclusion, A = O or B = O, is false.
So the given conditional statement is false.
So A = [0 1 / 0 0], B = [1 0 / 0 0]
is the counterexample.
So
A = [0 1 / 0 0]
B = [1 0 / 0 0]
is the answer.
Property 2
Property
If AB = O,
then BA = O.
This statement is false.
Recall that
AB and BA are not always equal.
Multiply Matrices
So BA is not always a zero matrix.
Example 2
Example
Solution
Again, a counterexample is an example
that makes the statement false.
And the given statement is a conditional statement.
So, set a counterexample
that seems to make
the hypothesis, AB = O, true
and the conclusion, BA = O, false.
Set A = [0 0 / 1 0] and B = [0 0 / 0 1].
First, show that
AB = O is true.
A = [0 0 / 1 0]
B = [0 0 / 0 1]
So AB = [0 0 / 1 0][0 0 / 0 1].
Solve [0 0 / 1 0][0 0 / 0 1].
Row 1, column 1:
0⋅0 + 0⋅0
Row 1, column 2:
0⋅0 + 0⋅1
Row 2, column 1:
1⋅0 + 0⋅0
Row 2, column 2:
1⋅0 + 0⋅1
So
[0 0 / 1 0][0 0 / 0 1]
= [0⋅0 + 0⋅0 0⋅0 + 0⋅1 / 1⋅0 + 0⋅0 1⋅0 + 0⋅1].
0⋅0 + 0⋅0
= 0 + 0
0⋅0 + 0⋅1
= 0 + 0
1⋅0 + 0⋅0
= 0 + 0
1⋅0 + 0⋅1
= 0 + 0
0 + 0 = 0
The elements of the matrix is all 0.
So this matrix is a zero matrix.
You got AB = O.
So the given hypothesis, AB = O,
is true.
Next, show that
BA = O is false.
A = [0 0 / 1 0]
B = [0 0 / 0 1]
So BA = [0 0 / 0 1][0 0 / 1 0].
Solve [0 0 / 0 1][0 0 / 1 0].
Row 1, column 1:
0⋅0 + 0⋅1
Row 1, column 2:
0⋅0 + 0⋅0
Row 2, column 1:
0⋅0 + 1⋅1
Row 2, column 2:
0⋅0 + 1⋅0
So
[0 0 / 0 1][0 0 / 1 0]
= [0⋅0 + 0⋅1 0⋅0 + 0⋅0 / 0⋅0 + 1⋅1 0⋅0 + 1⋅0].
0⋅0 + 0⋅1
= 0 + 0
0⋅0 + 0⋅0
= 0 + 0
0⋅0 + 1⋅1
= 0 + 1
0⋅0 + 1⋅0
= 0 + 0
0 + 0 = 0
0 + 1 = 1
[0 0 / 1 0] is not a zero matrix.
So
[0 0 / 1 0] ≠ O.
You got BA ≠ O.
So the given conclusion, BA = O,
is false.
See the given conditional statement.
The hypothesis, AB = O, is true.
The conclusion, BA = O, is false.
So the given conditional statement is false.
So A = [0 0 / 1 0], B = [0 0 / 0 1]
is the counterexample.
So
A = [0 0 / 1 0]
B = [0 0 / 0 1]
is the answer.