# 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**.

AB ≠ BA

**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.