Let’s look at some properties of multiplication of matrices.
1. Commutativity is not true:
AB ≠ BA
2. Zero matrix on multiplication
If AB = O,
then A ≠ O, B ≠ O is possible
3. Associative law:
(AB) C = A (BC)
4. Distributive law:
A (B + C) = AB + AC
(A + B) C = AC + BC
5. Multiplicative identity:
For a square matrix A
AI = IA = A
where I is the identity matrix of the same order as A.
Let’s look at them in detail
We used these matrices
Commutativity in multiplication is not true
AB ≠ BA
Let’s solve them
AB
BA
Since
∴ AB ≠ BA
Zero matrix multiplication
We saw that
So, AB = O
But A ≠ O & B ≠ O
Therefore,
If two matrices multiply to become zero matrix,
then it is not true that A = O or B = O
Note: This is different from numbers
If ab = 0,
then either a = 0 or b = 0But this is not true for matrices
Associative law
(AB) C = A (BC)
Let’s solve this
(AB) C
Note: Any matrix multiplied to zero matrix is a zero matrix
(AB) C = O × C
= O
A (BC)
.
Therefore,
(AB) C = A (BC)
Distributive law
Distributive law says that 
 A (B + C) = AB + AC

(A + B) C = AC + BC
Let’s prove both of them
A (B + C) = AB + AC
AB + AC
Therefore,
A (B + C) = AB + AC
Let’s prove the next one
(A + B) C = AC + BC
Therefore,
(A + B) C = AC + BC
Multiplicative Identity
For any square matrix A,
AI = IA = A
Where I is identity matrix of same order as A
Therefore,
AI = IA = A
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