## 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 2. Matrix Inverse Calculator - 2x2 Matrix

3. Matrix Inverse Calculator - 3x3 Matrix

4. Matrix Inverse Calculator - 4x4 Matrix

5. Cramer's Rule Calculator - 3x3 Matrix

6. Matrix Addition Calculator - 3x3 Matrix

7. Matrix Subtraction Calculator - 3x3 Matrix

8. Matrix Multiplication Calculator - 2x2 Matrix

9. Matrix Multiplication Calculator - 3x3 Matrix

10. Matrix Determinant Calculator - 3x3 & 2x2 Matrix

11. Matrix Addition Calculator - 2x2 Matrix

12. Matrix Subtraction Calculator- 2x2 Matrix

13. Matrix Addition Calculator - 4x4 Matrix

14. Matrix Subtraction Calculator- 4x4 Matrix

15. Matrix Multiplication Calculator - 4x4 Matrix

16. Matrix Determinant Calculator - 4x4 matrix

17. Squared Matrix Calculator

18. Transpose Matrix Calculator