Algebraic Properties of Matrices

1. Matrix addition is commutative.

If A and B are any two matrices of the same order, then

A + B = B + A

2. Matrix addition is associative.

If A, B and C are any three matrices of the same order then,

A+(B+C) = (A+B)+C

3. Additive identity

Let A be any matrix, then

A + O = O + A = A

This property is known as identity property of matrix addition. The zero matrix O is known as the identity element with respect to matrix addition.

4. Additive inverse

Let A be any matrix, then

A + (−A) = (−A) + A = O

This property is known as inverse property with respect to matrix addition. The negative of matrix A i.e. −A is the inverse of A with respect to matrix addition. 

5. Matrix multiplication is not commutative.


6. Matrix multiplication is associative

A(BC) = (AB)C 

7. Matrix multiplication is distributive over addition

A(B + C) = AB + AC

(A + B)C = AC + BC

8. Identity property of matrix multiplication

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