1. Introduction to Properties of Matrix Subtraction
Matrix subtraction behaves in predictable ways, similar to ordinary arithmetic, but with one important requirement — the matrices involved must be of the same order. Understanding these properties helps simplify expressions and solve matrix equations smoothly.
2. Property 1: Not Commutative
Matrix subtraction is not commutative. That means:
\( A - B \neq B - A \)
Swapping the order changes the result.
2.1. Example
A = \begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 5 \\ 0 & 2 \end{bmatrix}
Then:
A - B = \begin{bmatrix} 3 & -3 \\ 1 & 1 \end{bmatrix}
B - A = \begin{bmatrix} -3 & 3 \\ -1 & -1 \end{bmatrix}
The results are clearly different.
3. Property 2: Not Associative
Matrix subtraction is not associative. This means:
\( (A - B) - C \neq A - (B - C) \)
The order in which you group the matrices matters.
3.1. Example
A = \begin{bmatrix} 6 & 3 \end{bmatrix}, B = \begin{bmatrix} 2 & 1 \end{bmatrix}, C = \begin{bmatrix} 1 & -2 \end{bmatrix}
Evaluating both ways gives different answers.
4. Property 3: Existence of Additive Identity
The zero matrix acts as the identity for subtraction:
\( A - O = A \)
Subtracting a zero matrix does not change the matrix.
4.1. Example
A = \begin{bmatrix} 5 & -1 \\ 2 & 3 \end{bmatrix}, O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
Then:
A - O = A
5. Property 4: Additive Inverse
For every matrix \( A \), there exists an additive inverse \( -A \) such that:
\( A - A = O \)
This follows because subtracting a matrix from itself gives the zero matrix.
5.1. Example
A = \begin{bmatrix} 3 & -2 \\ 1 & 4 \end{bmatrix}
Then:
A - A = O
6. Property 5: Closure Property
If two matrices have the same order, their difference is also a matrix of the same order. Matrix subtraction is closed within matrices of fixed size.
Example: Subtracting two \( 3 \times 3 \) matrices always gives another \( 3 \times 3 \) matrix.
7. All Properties at a Glance
- Not commutative: \( A - B \neq B - A \)
- Not associative: \( (A - B) - C \neq A - (B - C) \)
- Identity element: \( A - O = A \)
- Additive inverse: \( A - A = O \)
- Closure: Difference of same-order matrices has the same order
These properties help you understand how matrix subtraction behaves and how to manipulate expressions safely.