Matrix Addition 3×3
Online calculator for adding two 3x3 matrices
Matrix Addition Calculator
Instructions
Enter the values of both matrices to be added. Empty fields are counted as zero. Click on Calculate.
Matrix Addition - Overview
Requirements
For matrix addition, the matrices must match. That is, they must have the same number of rows and columns.
Example
Given:
\(\displaystyle\begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix} + \begin{bmatrix}1 & 4 & 7\\2 & 5 & 8\\3 & 6 & 9\end{bmatrix} \)
Calculation:
\( = \begin{bmatrix}1+1 & 2+4 & 3+7\\4+2 & 5+5 & 6+8\\7+3 & 8+6 & 9+9\end{bmatrix} \)
Result:
\( = \begin{bmatrix}2 & 6 & 10\\6 & 10 & 14\\10 & 14 & 18\end{bmatrix} \)
Properties
- Commutative: A + B = B + A
- Associative: (A + B) + C = A + (B + C)
- Zero element: A + 0 = A
- Inverse: A + (−A) = 0 (zero matrix)
- Distributive: k(A + B) = kA + kB
Description of Matrix Addition
Basics
For matrix addition, the matrices must match. That is, they must have the same number of rows and columns. In matrix addition, the individual elements of the matrices are added together.
General Formula:
\(\displaystyle\begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{bmatrix} + \begin{bmatrix}b_{11} & b_{12} & b_{13}\\b_{21} & b_{22} & b_{23}\\b_{31} & b_{32} & b_{33}\end{bmatrix}\)
\(=\begin{bmatrix}a_{11}+b_{11} & a_{12}+b_{12} & a_{13}+b_{13}\\a_{21}+b_{21} & a_{22}+b_{22} & a_{23}+b_{23}\\a_{31}+b_{31} & a_{32}+b_{32} & a_{33}+b_{33}\end{bmatrix} \)
Calculation Rules
- Element-wise: Each element is added individually
- Same dimension: Both matrices must be the same size
- Commutative: A + B = B + A (order doesn't matter)
- Zero matrix: A + 0 = A (neutral element)
Detailed Example
Step-by-Step Calculation
Problem:
\(\displaystyle\begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix} + \begin{bmatrix}1 & 4 & 7\\2 & 5 & 8\\3 & 6 & 9\end{bmatrix} \)
Step 1: Add element-wise
\( = \begin{bmatrix}1+1 & 2+4 & 3+7\\4+2 & 5+5 & 6+8\\7+3 & 8+6 & 9+9\end{bmatrix} \)
Step 2: Result
\( = \begin{bmatrix}2 & 6 & 10\\6 & 10 & 14\\10 & 14 & 18\end{bmatrix} \)
Properties
- Commutative: A + B = B + A
- Associative: (A + B) + C = A + (B + C)
- Distributive: k(A + B) = kA + kB
- Zero element: A + 0 = A
- Additive inverse: A + (−A) = 0
- Identity: A + 0 = 0 + A = A
Practical Applications
Mathematics & Physics:
- Combining linear transformations
- Adding measurements and values
- State transitions in systems
Computer Science & Engineering:
- Image processing (overlay, brightness)
- Neural networks (combining layers)
- Data analysis (summing datasets)
Important Note
Matrix addition is commutative! This means: A + B = B + A. The order of the matrices does not matter. This is a fundamental difference from matrix multiplication, where order is important.