1. Consider the following matrices:
A = (
1
2
1
0
3
); F=(1 2 3)
2 1); B=(
2 1 1
@
1
2
A
@
1
0
3
A
@
1
A
2
1
2
1 1
1
C = 0
3
1
1
; D=0
1
1
2
1; E=0
2
1
Compute the following expressions if de ned: AB, BA, D2, B2, DC, CB,
BC, FE, EF, CE, EC.
2. Consider the following matrices:
; C=(
1
2
1
3
2
3
A=(2 1); B=(
2 6
)
1
2 )
@
1
1
0
A
@
3
0
2
A
1
0
1
1
2
6
D = 0
0
1
1
1; E=0
2
1 2
1 :
For each one of these matrices determine whether it is invertible or not, and compute its inverse when relevant.
Find x; y 2 R which solve the equation
A
( y )
= (
12
)
:
x
iii. Find x; y 2 R which solve the equation
B
( y )
= (
12
)
:
x
iv. Find a matrix G 2 M3(R) which solves the equation
DG=E:
3. In each of the following there is a claim, which might be true or false. If the claim is true then prove it, and if it is false then provide a counterexample. (For counter examples you may choose any n you wish, but if you want to
prove a claim then you should prove it for all possible n's).
If A 2 Mn(R) satis es A2 = 0 then A = 0. (Here 0 is the zero matrix).
If A; B 2 Mn(R) are such that AB = BA then AB2 = B2A.
Let A; B; C 2 Mn(R). If AB = CB then A = C.
1
2
In each of the following there is a claim, which might be true or false. If the claim is true then prove it, and if it is false then provide a counterexample. (For counter examples you may choose any n you wish, but if you want to prove a claim then you should prove it for all possible n's).
d. If A; B 2 Mn(R) are both invertible then AB is also invertible and (AB) 1=B 1A 1.
If A; B 2 Mn(R) are such that AB is invertible then A and B are also invertible.
e. If A; B 2 Mn(R) are such that A + B is invertible then A and B are both invertible.
e. If A; B 2 Mn(R) are both invertible then A + B is also invertible.
d. If A; B 2 Mn(R) are such that AB is invertible then BA is also invert-ible.
g. If A 2 Mn(R) is such that A3 is invertible, then A is invertible.
Prove the following claims.
i A square matrix A 2 Mn(R) is called 'diagonal' if all of its entries that are not on the main diagonal are equal zero, that is, A is diagonal if
(A)ij = 0 for all i ≠ j. Here is an example of a diagonal matrix:
1
3 0 0
@
0
0
1
A :
0
2
0
Prove that if A; B 2 Mn(R) are both diagonal then both A + B and AB are diagonal as well.
ii. For a square matrix A 2 Mn(R) the 'trace' of A, denoted tr(An), is the
Here is an example of a trace computation:
tr(A) = ∑i=1(A)ii.
sum of all of its entries on the main diagonal, that is
@
3
5
7
A
8
1
1
tr 0
0
2
4
1=3+2+( 1)=4:
For A; B 2 Mn(R) prove that tr(AB) = tr(BA).
For a square matrix A 2 Mn(R) the 'transposed' of A, denoted AT , is the matrix obtained by turning each row of A into a column by order,
that is (AT )i;j = (A)j;i. Here is an example of a transposed computation:
0 0
2
4
1
= 0
5
2
1
1 :
3
5
7
T
3
0
8
@
A
@
A
8
1
1
7
4
1
For A; B 2 Mn(R) prove that (AB)T = BT AT .
