Multiplication Properties of Matrices





In this page multiplication properties of matrices we are going to see some properties in multiplication.

Matrix Multiplication is not commutative:

For any two matrices multiplication is not commutative. That is AB ≠ BA.

Example 1:

A =
 
1 2
5 3
 
 
B =
 
2 5
7 3
 
AB=
 
1 2
5 3
 
x
 
2 5
7 3
 
 


=
 
1 2
 
x
 
2
7
 
 
 
1 2
 
x
 
5
3
 


  
 
5 3
 
x
 
2
7
 
 
 
5 3
 
x
 
5
3
 

=
 
(2+14) (5+6)
(10+21) (25+9)
 

AB=
 
16 11
31 34
 

Now let us find BA

BA=
 
2 5
7 3
 
x
 
1 2
5 3
 
 


=
 
2 5
 
x
 
1
5
 
 
 
2 5
 
x
 
2
3
 


  
 
7 3
 
x
 
1
5
 
 
 
7 3
 
x
 
2
3
 

=
 
(2+25) (4+15)
(7+15) (14+9)
 

BA=
 
27 19
22 23
 

From the above example we come to know that is AB ≠ BA.

Matrix Multiplication is Associative:

Matrix multiplication is associative A(BC) = (AB)C

A =
 
1 2
5 3
 
 
B =
 
2 5
7 3
 
C =
 
1 3
5 1
 

First we have to find BC

BC=
 
2 5
7 3
 
x
 
1 3
5 1
 
 


=
 
2 5
 
x
 
1
5
 
 
 
2 5
 
x
 
3
1
 


  
 
7 3
 
x
 
1
5
 
 
 
7 3
 
x
 
3
1
 

=
 
(2+25) (6+5)
(7+15) (21+3)
 

=
 
27 11
22 24
 
A(BC)=
 
1 2
5 3
 
x
 
27 11
22 24
 
 

=
 
(27+44) (11+48)
(135+66) (55+72)
 

=
 
71 59
201 127
 

Now we have to find AB

AB=
 
1 2
5 3
 
x
 
2 5
7 3
 
 

=
 
1 2
 
x
 
2
7
 
 
 
1 2
 
x
 
5
3
 


  
 
5 3
 
x
 
2
7
 
 
 
5 3
 
x
 
5
3
 

=
 
(2+14) (5+6)
(10+21) (25+9)
 

=
 
16 11
31 34
 


(AB)C=
 
16 11
31 34
 
x
 
1 3
5 1
 
 


=
 
16 11
 
x
 
1
5
 
 
 
16 11
 
x
 
3
1
 


  
 
31 34
 
x
 
1
5
 
 
 
31 34
 
x
 
3
1
 

=
 
(16+55) (48+11)
(31+170) (93+34)
 

=
 
71 59
201 127
 

Matrix Multiplication is distributive over addition:

A(B+C)= AB + AC

(A+B)C= AC+AB

A =
 
1 2
5 3
 
 
B =
 
2 5
7 3
 
C =
 
1 3
5 1
 


B+C=
 
2 5
7 3
 
+
 
1 3
5 1
 
 

=
 
3 8
12 4
 


A(B+C)=
 
1 2
5 3
 
x
 
3 8
12 4
 
 


=
 
1 2
 
x
 
3
12
 
 
 
1 2
 
x
 
8
4
 


  
 
5 3
 
x
 
3
12
 
 
 
5 3
 
x
 
8
4
 

=
 
(3+24) (8+8)
(15+36) (40+12)
 

=
 
27 16
51 52
 
AB=
 
1 2
5 3
 
x
 
2 5
7 3
 
 

=
 
1 2
 
x
 
2
7
 
 
 
1 2
 
x
 
5
3
 


  
 
5 3
 
x
 
2
7
 
 
 
5 3
 
x
 
5
3
 

=
 
(2+14) (5+6)
(10+21) (25+9)
 

=
 
16 11
31 34
 

we have to find AC

AC=
 
1 2
5 3
 
x
 
1 3
5 1
 
 

=
 
1 2
 
x
 
1
5
 
 
 
1 2
 
x
 
3
1
 


  
 
5 3
 
x
 
1
5
 
 
 
5 3
 
x
 
3
1
 

=
 
(1+10) (3+2)
(5+15) (15+3)
 

=
 
11 5
20 18
 
AB+AC=
 
16 11
31 34
 
+
 
11 5
20 18
 
 

AB+AC=
 
(16+11) (11+5)
(31+20) (34+18)
 

AB+AC=
 
27 16
51 52
 

AI = IA = A where I is the unit matrix or identity matrix.

A =
 
1 2
5 3
 
 
I =
 
1 0
0 1
 
 
AI=
 
1 2
5 3
 
x
 
1 0
0 1
 
 

=
 
1 2
 
x
 
1
0
 
 
 
1 2
 
x
 
0
1
 


  
 
5 3
 
x
 
1
0
 
 
 
5 3
 
x
 
0
1
 

=
 
(1+0) (0+2)
(5+0) (0+3)
 

=
 
1 2
5 3
 

Now let us find IA    multiplication properties of matrices.

AI=
 
1 0
0 1
 
x
 
1 2
5 3
 
 

=
 
1 0
 
x
 
1
5
 
 
 
1 0
 
x
 
2
3
 

=
 
0 1
 
x
 
1
5
 
 
 
0 1
 
x
 
2
3
 

=
 
(1+0) (2+0)
(0+5) (0+3)
 

=
 
1 2
5 3
 

These are the properties in the topic multiplication properties of matrices. multiplication properties of matrices







Multiplication Properties of Matrices to Addition Properties
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