1. What Is Matrix Addition?
Matrix addition is the process of adding two matrices by adding their corresponding elements. The idea is simple: add entry to entry, position to position.
If you know how to add numbers in a table, you already understand matrix addition—the structure is the same.
2. Condition for Adding Two Matrices
You can add two matrices only if they have the same order. This means both matrices must have:
- the same number of rows
- the same number of columns
If their sizes do not match, addition is not possible.
\( A + B \text{ is defined only if } A \text{ and } B \text{ are both } m \times n. \)
3. Rule for Matrix Addition
If \( A = [a_{ij}] \) and \( B = [b_{ij}] \) are two matrices of the same order, then their sum \( A + B \) is the matrix obtained by adding corresponding entries:
\( (A + B)_{ij} = a_{ij} + b_{ij} \)
This rule applies to every position in the matrix.
4. Example: Adding Two Matrices
Let:
\( A = \begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 0 \\ -2 & 1 \end{bmatrix} \)
Since both are \( 2 \times 2 \) matrices, we can add them.
4.1. Adding Corresponding Elements
- \( 2 + 5 = 7 \)
- \( -1 + 0 = -1 \)
- \( 4 + (-2) = 2 \)
- \( 3 + 1 = 4 \)
4.2. Final Result
\( A + B = \begin{bmatrix} 7 & -1 \\ 2 & 4 \end{bmatrix} \)
5. Another Example
Add the matrices:
\( X = \begin{bmatrix} 1 & 4 & 2 \\ 0 & -3 & 5 \end{bmatrix}, \quad Y = \begin{bmatrix} 2 & -1 & 3 \\ 7 & 6 & -2 \end{bmatrix} \)
5.1. Corresponding Sums
Add position by position:
- \( 1 + 2 = 3 \)
- \( 4 + (-1) = 3 \)
- \( 2 + 3 = 5 \)
- \( 0 + 7 = 7 \)
- \( -3 + 6 = 3 \)
- \( 5 + (-2) = 3 \)
5.2. Final Matrix
\( X + Y = \begin{bmatrix} 3 & 3 & 5 \\ 7 & 3 & 3 \end{bmatrix} \)
6. Observations About Matrix Addition
Matrix addition behaves similarly to normal number addition:
- Commutative: \( A + B = B + A \)
- Associative: \( (A + B) + C = A + (B + C) \)
- Additive Identity: \( A + O = A \)
These properties make matrix operations easier to handle as you progress to more advanced concepts.