Answers to ‘some questions possibly relevant to MATA33’
If you spot a typo please email [email protected]
B.
When giving examples I tend to use
O, I,
1
×
2, or 2
×
2 matrices because they are simple and
faster to check with. To understand and relate the various matrix properties to each other, you have
to (i) get the definitions right, (ii) get the rules/thms/properties right, and (iii) you have to use sound
reasoning.
1. True. You can always reduce a matrix; whether it reduces to
I
is another question.
2. Sometimes true, e.g.
A
=
I
. Sometimes false, e.g.
A
=
O
. Remember that a lower triangular
matrix does not need to have all nonzero entries in its main diagonal.
3. True, in order for the matrix multiplication in
AA

1
=
I
n
to be valid.
4. True.
A
is
m
×
n
for some natural numbers
m
and
n
. For
A
2
=
A
m
×
n
A
m
×
n
to be a matrix,
m
=
n
must be true. The converse (if
A
is
n
×
n
then
A
2
exists) is also true.
5. True. The matrix multiplication here always involves an
n
×
n
matrix multiplied by another
n
×
n
matrix.
6. Sometimes true, e.g.
A
=
I
. Sometimes false, e.g.
A
=
O
. Remember that a diagonal matrix
does not need to have all nonzero entries in its main diagonal.
7. True. There is no leading one in the zerorow, which implies

A

= 0, which implies
A
is not
invertible.
8. Sometimes, because I did not specify
A
to be square.
If
A
is not square, it may or may not reduce to a matrix that has no zerorow, e.g. [11] or [10].
If
A
is square, the statement is true. Any matrix that is invertible reduces to
I
, so any matrix
that is not invertible, the only other possibility, does not reduce to
I
. Any square matrix that
does not reduce to
I
must have a zerorow; otherwise it would have
n
leading ones its
n
rows,
which would have to be
I
.
9. Sometimes true, e.g.
A
=
B
=
I
. Sometimes false, e.g.
A
and
B
are any two invertible
n
×
n
matrices (they both reduce to
I
n
).
10. True. Any time a question asks you about a matrix and its inverse and actually gives you what
they could be, just multiply the matrix with its inverse (in either order) and see if you get
I
.
In this case, we have the matrix
A

1
and its supposed inverse
A
. Well,
A

1
A
=
I
because
A
is
invertible. In a sense you are just reading the equation backwards.
11. True, see above. Just invert
A

1
in whichever way you like to get
A
.
12. True. Use the property (
XY
)
T
=
Y
T
X
T
. We have
(
ABC
)
T
= [
A
(
BC
)]
T
order of backets don’t matter in matrix mult.
= (
BC
)T
A
T
by the property
=
C
T
B
T
A
T
by the property
The example on p. 245 is more complicated because it takes the transpose of one of the matrices.
13. Sometimes true, e.g.
A
=
O
. Sometimes false, e.g.
A
=
0
1
0
0
. See Assignment 2 #5b.
14. Sometimes true, e.g. when
A
=
B
.
Sometimes false.
Matrix multiplication isn’t commutative, which just means that you can’t
always switch the order of them.
To make an example, plug in some random numbers and
insert some zeroes to make checking faster. For example,
A
=
0
1
0
0
and
B
=
1
0
0
0
.
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15. If
R
is a reduced matrix, then every entry of
R
is either 0 or 1.
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 Spring '11
 G
 Matrices, Diagonal matrix

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