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Linear Algebra
Grinshpan
Patterns of matrix multiplication
When the number of columns of a matrix A agrees with the number of rows of a matrix
B, we may speak of the product matrix AB. The (i, j)-entry of AB is obtained by
multiplying together ith row of A and jth column of B :
 
b
1j
..  = a b + . . . + a b .

(AB)ij = ai1 . . . ain
i1 1j
in nj
.
bnj
AB has as many rows as A and as many columns as B. The matrix product has many
properties and patterns. Some are listed below. These hardly require memorization, but do
require some recognition.
A row times a column is a number, but a column times a row is a full

 
 
xa xb
x
x
y  a b c = ya yb
a b c y  = ax + by + cz
za zb
z
z
grown matrix:

xc
yc
zc
Nonzero matrices may have zero product:
1 −1 1 1
0 0
0 1 0 1
0 0
=
=
−1 1
1 1
0 0
0 0 0 0
0 0
A zero

∗ ∗
∗ ∗
0 0
row of the

∗ ∗ ∗
∗ ∗ ∗
∗ ∗
0
A zero

∗ ∗
∗ ∗
∗ ∗
column

∗ ∗
∗ ∗
∗ ∗
first factor is
 
∗
∗ ∗


∗ = ∗ ∗
∗
0 0
inherited by the product:

∗
∗
0
of the second factor is inherited by the product:
 

0 ∗
∗ 0 ∗
0 ∗ = ∗ 0 ∗
0 ∗
∗ 0 ∗
The

a
0
0
product of diagonal matrices

 
0 0
x 0 0
ax 0




b 0
0 y 0 = 0 by
0 c
0 0 z
0 0
is again a diagonal matrix:

0
0
cz
The

∗
0
0
product of upper (lower) triangular matrices is again upper (lower) triangular:
 


∗ ∗ ∗ ∗ ∗
∗ ∗ ∗
∗ ∗  0 ∗ ∗ =  0 ∗ ∗
0 0 ∗
0 0 ∗
0 ∗
Multiplication


a 0 0
1
0 b 0 1
0 0 c
1
by a diagonal
 
1 1
a
1 1 =  b
1 1
c
matrix on the left has a scaling effect on rows:

a a
b b
c c
Multiplication


1 1 1 a
1 1 1 0
1 1 1
0
by a diagonal
 
0 0
a
b 0 = a
0 c
a
matrix on the right has a scaling effect on columns:

b c
b c
b c
Left multiplication by an identity matrix with switched rows switches rows:


 

0 1 0 a b c
d e f
1 0 0 d e f  = a b c 
0 0 1
g h i
g h i
Right multiplication by an identity matrix with switched columns switches columns:

 


b a c
0 1 0
a b c
d e f  1 0 0 =  e d f 
h g i
0 0 1
g h i
Left multiplication by a shear matrix has the corresponding effect on rows:

 


a+d b+e c+f
1 1 0 a b c
0 1 0 d e f  =  d
e
f 
add row 2 to row 1
g
h
i
g h i
0 0 1

 


a
b
c
1 0 0 a b c
 0 1 0 d e f  =  d
e
f 
subtract row 1 from row 3
−a + g −b + h −c + i
g h i
−1 0 1
Right multiplication by a shear matrix has the corresponding effect on columns:


 

a b c
1 1 0
a a+b c
d e f  0 1 0 = d d + e f 
add column 1 to column 2
g h i
0 0 1
g g+h i


 

a b c
1 0 0
a + 2c b c
d e f  0 1 0 = d + 2f e f 
add 2 columns 3 to column 1
g h i
2 0 1
g + 2i h i
Column-row expansion
If C1 , . . . , Cn are the columns of A and R1 , . . . , Rn are the rows of B, then
AB = C1 R1 + . . . + Cn Rn .
a b a b x y
ax + bz ay + bt
ax ay
bz bt
x y +
z t
=
=
=
+
c
d
c d z t
cx + dz cy + dt
cx cy
dz dt