## 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.

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

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

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

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. 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 