1. Introduction to Properties of Matrix Multiplication
Matrix multiplication does not follow all the rules of normal number multiplication. Some rules are the same, while others change completely. Understanding these properties helps you work confidently with matrix expressions and simplifies longer calculations.
Below are the most important properties, each described in simple language with examples.
2. Property 1: Non-Commutativity
Matrix multiplication is not commutative. In general:
\( AB \neq BA \)
Even when both products are defined, switching the order usually gives a different answer.
2.1. Example
Let:
A = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 1 \\ -2 & 5 \end{bmatrix}
Compute:
- \( AB = \begin{bmatrix} 0 & 11 \\ -6 & 15 \end{bmatrix} \)
- \( BA = \begin{bmatrix} 4 & 11 \\ -2 & 13 \end{bmatrix} \)
Clearly, \( AB \neq BA \).
3. Property 2: Associative Property
Matrix multiplication is associative. When multiplying three matrices, the grouping does not matter:
\( A(BC) = (AB)C \)
As long as all products involved are defined, the result is the same.
3.1. Example
If:
A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}, \; B = \begin{bmatrix} 3 & 1 \\ 4 & 2 \end{bmatrix}, \; C = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}
Then both \( A(BC) \) and \( (AB)C \) give the same product.
4. Property 3: Distributive Property
Matrix multiplication distributes over matrix addition:
\( A(B + C) = AB + AC \)
\( (A + B)C = AC + BC \)
This works just like the distributive law for real numbers.
4.1. Example
Let:
B = \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix}, \; C = \begin{bmatrix} 1 & 4 \\ 0 & 2 \end{bmatrix}
Then:
B + C = \begin{bmatrix} 3 & 5 \\ -1 & 5 \end{bmatrix}
Multiplying both sides by another matrix will confirm the law.
5. Property 4: Identity Matrix Property
The identity matrix behaves like the number 1 in multiplication:
\( AI = IA = A \)
This holds for any square matrix \( A \) of the same order as \( I \).
5.1. Example
If:
A = \begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix}, \; I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
Then:
AI = IA = A
6. Property 5: Zero Matrix Property
Multiplying any matrix by a zero matrix of compatible order gives the zero matrix.
\( AO = O \quad \text{and} \quad OB = O \)
Here, the sizes of the matrices must match the multiplication requirement.
6.1. Example
Let:
A = \begin{bmatrix} 3 & 1 \\ 2 & -4 \end{bmatrix}, \quad O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
Then:
AO = O
7. Property 6: Compatibility Requirement
Matrix multiplication is not always possible. You can multiply \( A_{m \times n} \) and \( B_{p \times q} \) only when:
\( n = p \)
If this condition fails, the product is not defined.
8. All Properties at a Glance
- Not commutative: \( AB \neq BA \)
- Associative: \( A(BC) = (AB)C \)
- Distributive: \( A(B + C) = AB + AC \)
- Identity: \( AI = IA = A \)
- Zero matrix: \( AO = OB = O \)
- Compatibility: multiplication defined only if columns of A = rows of B
These rules form the foundation of matrix algebra and are essential for solving matrix problems efficiently.