1. Introduction to Properties of Matrix Addition
Matrix addition follows rules that are very similar to the rules you already know from ordinary arithmetic. These properties make it easy to simplify expressions and work confidently with matrices during calculations.
Below are the main properties of matrix addition explained in simple language with examples.
2. Property 1: Commutative Property
The order of addition does not matter. You get the same result whether you add \( A + B \) or \( B + A \).
\( A + B = B + A \)
2.1. Example
Let:
A = \begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 1 \\ 0 & 2 \end{bmatrix}
Then:
A + B = \begin{bmatrix} 7 & 4 \\ -1 & 6 \end{bmatrix}
B + A = \begin{bmatrix} 7 & 4 \\ -1 & 6 \end{bmatrix}
Both are the same, showing commutativity.
3. Property 2: Associative Property
When adding three matrices, the grouping does not affect the result.
\( (A + B) + C = A + (B + C) \)
3.1. Example
Let:
A = \begin{bmatrix} 1 & 2 \end{bmatrix}, \; B = \begin{bmatrix} 3 & 4 \end{bmatrix}, \; C = \begin{bmatrix} 5 & 6 \end{bmatrix}
Both groupings will give:
\( \begin{bmatrix} 9 & 12 \end{bmatrix} \)
4. Property 3: Additive Identity
The zero matrix acts like 0 in normal arithmetic. Adding it to any matrix leaves the matrix unchanged.
\( A + O = A \)
4.1. Example
A = \begin{bmatrix} 4 & -1 \\ 2 & 3 \end{bmatrix}, \quad O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
Then:
A + O = A
5. Property 4: Additive Inverse
For every matrix \( A \), there exists a matrix \( -A \) such that:
\( A + (-A) = O \)
The matrix \( -A \) is obtained by changing the sign of every entry of \( A \).
5.1. Example
A = \begin{bmatrix} 3 & -2 \\ 1 & 5 \end{bmatrix}
Then:
-A = \begin{bmatrix} -3 & 2 \\ -1 & -5 \end{bmatrix}
Adding them gives the zero matrix.
6. Property 5: Closure Property
If two matrices have the same order, their sum is also a matrix of the same order. Matrix addition is always closed within the set of matrices of a fixed size.
Example: Adding two \( 2 \times 3 \) matrices always gives another \( 2 \times 3 \) matrix.
7. All Properties at a Glance
- Commutative: \( A + B = B + A \)
- Associative: \( (A + B) + C = A + (B + C) \)
- Additive Identity: \( A + O = A \)
- Additive Inverse: \( A + (-A) = O \)
- Closure: Sum of two same-order matrices is also the same order.
These rules make matrix addition predictable and easy to use in larger computations.