Download Definition Let A be an m × n matrix and B an n × p matrix. Then, their

Definition Let A be an m × n matrix and B an n × p matrix. Then,
their product AB is the m × p matrix whose (i, k)-th entry is the
row–column product of the ith row of A with the kth column of B.
Example Let A =
1 2
3 4
(AB)11 =
(AB)12
=
(AB)21 =
(AB)22
so A B =
BA=
3 7
11 15
5 1
−1 3
!
=
!
1 2
3 4
5 1
−1 3
and B =
1 2
!
3 4
5
−1
1
3
!
5
−1
1
3
!
!
!
!
. Then,
=5−2=3
=1+6=7
= 15 − 4 = 11
= 3 + 12 = 15
. On the other hand,
1 2
3 4
!
=
5+3
10 + 4
−1 + 9 −2 + 12
!
=
8 14
8 10
!
.
Example
Let
A =
1 3 2
2 1 4
!
, B =
3 1
−1 0
!
.
Then A is a 2 × 3 matrix, and B is a 2 × 2 matrix. Since the dimensions do not agree in the required way, the product AB does not
exist.
Example Let A and B be as in the previous example. If we let m = 2,
n = 2 and p = 3 then B is an m × n matrix and A in an n × p matrix,
so we can define the product B A, and we expect an m × p matrix
(2 × 3) for the product:
BA =
=
=
3 1
−1 0
!
1 3 2
2 1 4
!
3+2 9+1 6+4
−1
−3
−2
5 10 10
−1 −3 −2
!
.
!
Let A =
1 2
3 4
(i) AB =
!
1 −1 1
0
2 3
,B=
1 3 7
3 5 15
!


1


and C =  4 . Then
3
!
;
(ii) AC is not defined, due to incompatible dimensions;
(iii) BC =
(iv) ABC =
0
17
!
34
68
;
!
;
(v) BA is not defined, due to incompatible dimensions;


1 2


(vi) B T A =  5 6 .
10 14
Remark. The product formula can be written reasonably compactly:
if A = (aij ) is m × n and B = (bjk ) is n × p (so number of columns
of A = number of rows of B) then we define
AB = (aij )(bjk ) = (cik ) where cik =
n
X
j=1
aij bjk .
Properties of matrix multiplication We have associative and distributive laws for matrices. Assuming the indicated operations can be
performed on matrices A, B, C, then:
1. (AB)C = A(BC)
2. A(B + C) = AB + AC
3. (A + B)C = AC + BC
4. α(AB) = (αA)B = A(αB), α ∈ R.
If A, B are square of the same size, then
(A + B)2 = (A + B)(A + B)
= A(A + B) + B(A + B)
= A2 + AB + BA + B 2.
We cannot assume AB = BA though, recall example above!!
If A and B are such that their product is defined, then
(A B)T = B T AT .
For every integer n > 0, the n × n identity matrix In “fixes” a matrix
under multiplication. (Recall, In is diagonal and has all its diagonal
entries equal to 1.) For example,
I2 =
1 0
0 1
!
,


1 0 0


I3 =  0 1 0  .
0 0 1
Basic property: if A is m × n, then
A In = Im A = A.
Some odd things
• Not all pairs of matrices can be multiplied together; the dimensions must be compatible.
• Generally, AB 6= BA, even if both products are defined and
have the same sizes (for example if A, B are both n × n). We
saw an example of this above. Here is another one:
1 2
0 1
!
2 1
−1 1
!
=
0 3
−1 1
!
,
whereas
2 1
−1 1
!
1 2
0 1
!
=
2
5
−1 −1
!
.
• Two non-zero matrices can multiply to give a zero matrix. For
example,
1
1
−1 −1
!2
=
0 0
0 0
!
.