Properties of
Matrix Operations
Discover why matrix multiplication breaks commutativity but preserves associativity and distributivity: the rules that govern how matrices combine.
What Is a Matrix?
A matrix is a rectangular array of numbers arranged in rows and columns. We describe its size as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.
Three Core Operations
Why Do Properties Matter?
When you solve systems of equations, transform graphics in video games, or train machine-learning models, you rearrange matrix expressions constantly. Knowing which rearrangements are valid prevents costly errors.
The Three Key Properties
Why \( AB \neq BA \)
Each entry of a product depends on a specific row from the left matrix and a specific column from the right matrix. Swapping the order feeds different rows into different columns, producing a completely different grid of dot products.
Associativity: Regrouping Is Safe
For any conformable matrices \( A \), \( B \), and \( C \):
This property lets you choose the most efficient grouping for computation without worrying about getting a different answer.
Distributive Property
Matrix multiplication distributes over addition from both sides:
Real Numbers vs Matrices
| Property | Real Numbers | Square Matrices |
|---|---|---|
| Commutativity | ✓ \( ab = ba \) | ✕ \( AB \neq BA \) in general |
| Associativity | ✓ \( (ab)c = a(bc) \) | ✓ \( (AB)C = A(BC) \) |
| Distributivity | ✓ \( a(b+c) = ab+ac \) | ✓ \( A(B+C) = AB+AC \) |
| Identity Element | ✓ \( a \cdot 1 = a \) | ✓ \( AI = IA = A \) |
| Zero Product | \( ab = 0 \Rightarrow a=0 \) or \( b=0 \) | ✕ \( AB = O \) possible with \( A,B \neq O \) |
How to Apply the Properties
Example 1: Non-Commutativity
Let \( A = \begin{bmatrix}1 & 0\\2 & 1\end{bmatrix} \) and \( B = \begin{bmatrix}0 & 1\\3 & 0\end{bmatrix} \). Compute \( AB \) and \( BA \).
\( = \begin{bmatrix}0 & 1\\3 & 2\end{bmatrix} \)
\( = \begin{bmatrix}2 & 1\\3 & 0\end{bmatrix} \)
Example 2: Verifying Associativity
Let \( A = \begin{bmatrix}1&2\\0&1\end{bmatrix}\), \( B = \begin{bmatrix}2&0\\1&3\end{bmatrix}\), \( C = \begin{bmatrix}1&1\\0&2\end{bmatrix}\). Show \( (AB)C = A(BC) \).
Example 3: Distributive Property
Let \( A = \begin{bmatrix}2&1\\0&3\end{bmatrix}\), \( B = \begin{bmatrix}1&0\\2&1\end{bmatrix}\), \( C = \begin{bmatrix}0&3\\1&0\end{bmatrix}\). Verify \( A(B+C) = AB + AC \).
The Identity Matrix \( I \)
The identity matrix has 1s on the main diagonal and 0s everywhere else. It is the multiplicative identity for matrices, just as 1 is for real numbers:
Common Mistakes to Avoid
Summary of Properties
| Property | Formula | Status |
|---|---|---|
| Commutativity | \( AB = BA \) | ✕ Fails in general |
| Associativity | \( (AB)C = A(BC) \) | ✓ Always holds |
| Left Distributive | \( A(B+C) = AB + AC \) | ✓ Always holds |
| Right Distributive | \( (B+C)A = BA + CA \) | ✓ Always holds |
| Identity | \( AI = IA = A \) | ✓ Always holds |
| Zero Product | \( AB = O \Rightarrow A=O \) or \( B=O \) | ✕ Fails in general |
Question 1 of 5
Which property does not hold for matrix multiplication in general?
Question 2 of 5
If \( A = \begin{bmatrix}1&2\\3&4\end{bmatrix} \), what is \( AI \) where \( I \) is the \( 2\times2 \) identity matrix?
Question 3 of 5
Which expression is a correct application of the distributive property?
Question 4 of 5
Given \( (AB)C = A(BC) \), which property is being demonstrated?
Question 5 of 5
If \( AB = O \) (the zero matrix), can we conclude that \( A = O \) or \( B = O \)?