# Matrices (Pt. 3): MM Properties

As the matrix multiplication is *that important*, it is only fitting that there are **3 basic properties** that govern this particular operation.

## Matrix Multiplication Properties

{% tabs %}
{% tab title="Not Commutative" %}

#### Multiplication of matrices are not commutative.

This means that:

$$
AB\neq BA
$$

Hence, positioning while performing multiplication of matrices do matter

Similarly, as would've expected, the number of columns (n) at left matrix should be the same as the number of rows at right matrix (m)
{% endtab %}

{% tab title="Associative" %}

#### Multiplication of matrices are associative.

This means that:

$$
(a \times b) \times c = a \times (b \times c)=a \times b \times c
$$

* Associative property is true if BODMAS rule (order of operations) does not matter  
* Hence, slotting in a parentheses at any pair of operands should yield the same final result

Do not confuse this with the **not commutative** property discussed in the previous tab, as the "raw form" of Matrix A should not interact with the "raw form" of Matrix C. 🥓
{% endtab %}

{% tab title="Distributive" %}
I was surprised when I saw this property. This seemed extremely straightforward, trivial, and even arguably pointless (lol) 🐱‍👓. However, I believe the main reason for spelling this as a property explicitly could be to instill a fundamental concept building towards more complicated problems.

#### Multiplication of matrices are distributive.

This means that:

$$
A (B+C) = AB+AC
$$

$$
(B+C)D = BC+BD
$$

With this property,

* While performing any superficial A multiply with B operation, we can subconsciously visualize either A or B being divided into smaller parts, **and**
* The latter matrix will be distributive to the division of each of its smaller parts
  {% endtab %}
  {% endtabs %}

#### **More on the next page ⏭**