Matrices

Definition. A 𝒎 × 𝒏 size matrix 𝑨 is a rectangular array of 𝑚 ∙ 𝑛 real (or

complex) numbers arranged in 𝑚 horizontal rows and 𝑛 vertical columns

𝐴 =

[

𝑎11 𝑎12⋯ 𝑎1𝑗 …𝑎1𝑛

𝑎21 𝑎22⋯ 𝑎2𝑗 …𝑎2𝑛

…………………… . .𝑎𝑖1 𝑎𝑖2⋯ 𝑎𝑖𝑗 …𝑎𝑖𝑛 ………………… . .

𝑎𝑚1 𝑎𝑚2⋯ 𝑎𝑚𝑗 …𝑎𝑚𝑛]

𝑖 𝑡ℎ 𝑟𝑜𝑤 (1)

j th column

We shall say that 𝐴 is 𝑚 by 𝑛 (written 𝑚 × 𝑛 ). If 𝑚 = 𝑛, we say that 𝐴 is

a square matrix of order 𝑛 and the numbers 𝑎11 , 𝑎22, … , 𝑎𝑛𝑛 are elements of

the main diagonal of 𝐴. We refer to the 𝑎𝑖𝑗, as 𝑖, 𝑗 th element of 𝐴 and we

often write (1) as

𝐴 = [𝑎𝑖𝑗].

Example 1. Let 𝐴 = [ 1 2 3−1 0 1

], 𝐵 = [1 42 −3

], 𝐶 = [ 1−1 2

], 𝐷 = [1 1 02 0 13 −1 2

].

Then 𝐴 is a 2 × 3 matrix with

𝑎11 = 1 𝑎12 = 2 𝑎13=3

𝑎21 = −1 𝑎22 = 0 𝑎23=1;

𝐵 is a 2 × 2 matrix with

𝑏11 = 1 𝑏12 = 4

𝑏21 = 2 𝑏22 = −3 ;

𝐶 is a 3 × 1 matrix; 𝐷 is a 3 × 3 matrix. In 𝐷, the elements 𝑑11 =

1, 𝑑22 = 0, 𝑑33 = 2 form the main diagonal.

Any 1 × 𝑛 size matrix is called row-vector, any 𝑛 × 1 size matrix is called

column-vector or simply are called a 𝑛 −vector.

Example 2. 𝑢 = [1 2 − 1 0 ] is 4-vector and 𝑣 = [ 1−1 3

] is 3-vector.

Definition. A square matrix 𝐴 = [𝑎𝑖𝑗] for which every term off the main

diagonal is zero, that is 𝑎𝑖𝑗 = 0 for 𝑖 ≠ 𝑗, is called diagonal matrix.

Example 3. 𝐺 = [ 4 0 0 1

], 𝐻 = [−3 0 00 −2 00 0 4

] are diagonal matrix.

Definition. A diagonal matrix 𝐴 = [𝑎𝑖𝑗] for which 𝑎𝑖𝑗 = 𝑐 for 𝑖 = 𝑗 and 𝑎𝑖𝑗 = 0

for ≠ 𝑗 , is called a scalar matrix.

Example 4. The following are scalar matrices: J = [ −2 0 0 − 2

],

I3 = [1 0 00 1 00 0 1

] .

Definition. Two 𝑚 × 𝑛 matrix 𝐴 = [𝑎𝑖𝑗] and 𝐵 = [𝑏𝑖𝑗] are said to be equal if

𝑎𝑖𝑗 = 𝑏𝑖𝑗 , that is if corresponding elements are equal.

Example 5. The matrices: A = [1 2 −12 −3 40 −4 5

] and B = [1 2 𝑤2 𝑥 4𝑦 −4 𝑧

] are

equal if 𝑤 = −1, 𝑥 = −3, 𝑦 = 0 𝑎𝑛𝑑 𝑧 = 5.

Definition. If 𝐴 = [𝑎𝑖𝑗] and 𝐵 = [𝑏𝑖𝑗] are 𝑚 × 𝑛 matrices, then the sum of 𝐴

and 𝐵 is the 𝑚 × 𝑛 matrix 𝐶 = [𝑐𝑖𝑗] , defined by

𝑐𝑖𝑗 = 𝑎𝑖𝑗 + 𝑏𝑖𝑗 .

That is, C is obtained by adding the corresponding elements of A and B.

Example 6. Let 𝐴 = [ 1 − 2 42 − 1 3

] and 𝐵 = [ 0 2 − 41 3 1

].

Then 𝐴 + 𝐵 = [ 1 + 0 − 2 + 2 4 + (−4)2 + 1 − 1 + 3 3 + 1

] = [1 0 0 3 2 4

].

Definition. If 𝐴 = [𝑎𝑖𝑗] is 𝑚 × 𝑛 matrix and 𝑟 is real number, then the scalar

multiple of 𝐴 by 𝑟 , 𝑟𝐴, is 𝑚 × 𝑛 matrix 𝐵 = [𝑏𝑖𝑗], where 𝑏𝑖𝑗 = 𝑟𝑎𝑖𝑗.

That is 𝐵 obtained by multiplying each element of 𝐴 by 𝑟.

If 𝐴 = [𝑎𝑖𝑗] and 𝐵 = [𝑏𝑖𝑗] are 𝑚 × 𝑛 matrices, we write 𝐴 + (−1)𝐵 as 𝐴 − 𝐵

and call this the difference of 𝐴 and 𝐵.

Example 7. Let 𝐴 = [ 2 3 − 54 2 1

] and 𝐵 = [2 − 1 33 5 − 2

].

Then 𝐴 − 𝐵 = [2 − 2 3 + 1 − 5 − 34 − 3 2 − 5 1 + 2

] = [0 4 − 81 − 3 3

].

Definition. If 𝐴 = [𝑎𝑖𝑗] is 𝑚 × 𝑛 matrix, then 𝑛 × 𝑚 matrix 𝐴𝑇 = [𝑎𝑖𝑗𝑇], where

𝑎𝑖𝑗𝑇 = 𝑎𝑗𝑖

is called the transpose of 𝐴.

Thus, the entries in each row of 𝐴𝑇 are the entries in the corresponding column

of 𝐴.

Example 8. Let 𝐴 = [4 − 2 0 0 5 − 2

] then 𝐴𝑇 = [4 0

−2 50 −2

],

𝐷 = [3 −5 1] then 𝐷𝑇 = [3

−51

].

Definition. If 𝐴 = [𝑎𝑖𝑗] is an 𝑚 × 𝑝 matrix and 𝐵 = [𝑏𝑖𝑗] is 𝑝 × 𝑛 matrix, then

the product of 𝐴 and 𝐵, denoted 𝐴𝐵, is the 𝑚 × 𝑛 matrix 𝐶 = [𝑐𝑖𝑗], defined

by

𝑐𝑖𝑗 = 𝑎𝑖1𝑏1𝑗 + 𝑎𝑖2𝑏2𝑗 + ⋯+ 𝑎𝑖𝑝𝑏𝑝𝑗 = ∑ 𝑎𝑖𝑘𝑏𝑘𝑗𝑝𝑘=1 .

j th column

𝑖 𝑡ℎ 𝑟𝑜𝑤

[ 𝑎11 𝑎12⋯ …… . 𝑎1𝑝

𝑎21 𝑎22 … …𝑎2𝑝

…………………… . .𝑎𝑖1 𝑎𝑖2⋯ ……𝑎𝑖𝑝 ………………… . .𝑎𝑚1 𝑎𝑚2⋯ …𝑎𝑚𝑝 ]

[ 𝑏11 𝑏12 …𝑏1𝑗 …𝑏1𝑛

𝑏21 𝑏22 …𝑏2𝑗 …𝑏2𝑛

…………………… . .…………………… . .………………… . .

𝑏𝑝1 𝑏𝑝2 …𝑏𝑝𝑗 …𝑏𝑝𝑛]

=

[

𝑐11 ……… 𝑐1𝑛

………… …𝑐𝑖1 ……𝑐𝑖𝑗 …𝑐𝑖𝑛

…………………𝑐𝑚1 ……𝑐𝑚𝑗 …𝑐𝑚𝑛]

Example 9. Let 𝐴 = [1 2 − 13 1 4

] and 𝐵 = [−2 54 −32 1

]. Then

𝐴𝐵 = [1 2 − 13 1 4

] [−2 54 −32 1

] =

[1 ∙ (−2) + 2 ∙ 4 + (−1) ∙ 2 1 ∙ 5 + 2 ∙ (−3) + (−1) ∙ 1

3 ∙ (−2) + 1 ∙ 4 + 4 ∙ 2 3 ∙ 5 + 1 ∙ (−3) + 4 ∙ 1]=[

4 −26 16

]

The Matrix Multiplication Is Not Commutative

Matrix multiplication is a noncommutative operation—i.e., it is possible

for AB = BA, even when both products exist and have the same shape.

Example 10. Let 𝐴 = [1 2

−1 3] and 𝐵 = [

2 10 1

]. Then 𝐴𝐵 = [2 3

−2 2] and

𝐵𝐴 = [2 10 1

] [1 2

−1 3] = [

1 7−1 3

].

Thus 𝐴𝐵 ≠ 𝐵𝐴.