Algebraic Properties of Matrices





In this page we are going to algebraic properties of matrices we are going to see some properties in the concept matrix.

Matrix Addition is Commutative:

If A and B are any two matrices of the same order then A+B = B+A. This property is known as commutative property of matrix addition.

A =
 
0 7
-3 11
 
 
B =
 
-3 1
-7 0
 

Now let us find A + B

A+B=
 
0 7
-3 11
 
+
 
-3 1
-7 0
 
 

A+B=
 
0+(-3) 7+1
-3+(-7) 11+0
 
 
A+B=
 
-3 8
-10 11
 
 

Now let us find B + A

B+A=
 
-3 1
-7 0
 
+
 
0 7
-3 11
 
 

B+A=
 
(-3)+0 1+7
(-7)+(-3) 0+11
 
 
B+A=
 
-3 8
-10 11
 
 

Matrix Addition is Associative:

If A,B and C are any three matrices of same order then A+(B+C) = (A+B)+C. This is the property of matrix addition.

A =
 
1 2
3 1
 
B =
 
2 3
7 6
 
C =
 
5 2
3 1
 
 

Now let us find B + C

B+C=
 
2 3
7 6
 
+
 
5 2
3 1
 
 

B+C=
 
(2+5) (3+2)
(7+3) (6+1)
 
 

B+C=
 
7 5
10 7
 
 

Now we have to find A + (B+C)

A+(B+C)=
 
1 2
3 1
 
+
 
7 5
10 7
 
 

A+(B+C)=
 
(1+7) (2+5)
(3+10) (1+7)
 
 

A+(B+C)=
 
8 7
13 8
 
 

Now let us find A + B

A+B=
 
1 2
3 1
 
+
 
2 3
7 6
 
 

Now we have to add corresponding terms


A+B=
 
(1+2) (2+3)
(3+7) (1+6)
 
 

A+B=
 
3 5
10 7
 
 

Now we have to add A+B with C.

(A+B)+C=
 
3 5
10 7
 
 
+
 
5 2
3 1
 
 

A+(B+C)=
 
(3+5) (5+2)
(10+3) (7+1)
 
 

A+(B+C)=
 
8 7
13 8
 
 

Additive Identity:

Let A be any matrix then A + O = 0 + A = A. This property i called additive property of matrix identity.

A =
 
1 2
3 5
 
O =
 
0 0
0 0
 

To check this,first we have to add A + O


A+O=
 
1 2
3 5
 
 
+
 
0 0
0 0
 
 

A+O=
 
(1+0) (2+0)
(3+0) (5+0)
 
 

A+O=
 
1 2
3 5
 
 

Now we have to add O+A


O+A=
 
0 0
0 0
 
 
+
 
1 2
3 5
 
 

O+A=
 
(0+1) (0+2)
(0+3) (0+5)
 
 

A+O=
 
1 2
3 5
 
 

Additive Inverse:

Let A be any matrix then A + (-A) = (-A) + A = o. This property is called as additive inverse. The zero matrix is also known as identity element with respect to matrix addition.

These are the properties in addition in the topic algebraic properties of matrices.







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