View source for Matrix Addition

= Matrix Addition =

Matrix addition is the process of adding two matrices of the '''same dimensions''' by adding their corresponding elements.

== Conditions for Matrix Addition ==

Two matrices can be added only if they have the same number of rows and the same number of columns.

For example, if:
* Matrix A is of order 2×3
* Matrix B must also be of order 2×3 to be added to A

== Rule for Addition ==

If:
<math>
A = [a_{ij}], \quad B = [b_{ij}]
</math>

Then:
<math>
A + B = [a_{ij} + b_{ij}]
</math>

This means we add each element of matrix A with the corresponding element in matrix B.

== Step-by-Step Example ==

Let:

<math>
A = \begin{bmatrix}
2 & 4 & 6 \\
1 & 3 & 5
\end{bmatrix}, \quad
B = \begin{bmatrix}
7 & 1 & 3 \\
4 & 0 & 2
\end{bmatrix}
</math>

'''Step 1:''' Check that both matrices have the same order.  
Both A and B are 2×3 matrices → ✅ OK

'''Step 2:''' Add corresponding elements:

<math>
A + B = \begin{bmatrix}
2+7 & 4+1 & 6+3 \\
1+4 & 3+0 & 5+2
\end{bmatrix}
</math>

'''Step 3:''' Perform the additions:

<math>
A + B = \begin{bmatrix}
9 & 5 & 9 \\
5 & 3 & 7
\end{bmatrix}
</math>

== Final Answer ==

<math>
\begin{bmatrix}
9 & 5 & 9 \\
5 & 3 & 7
\end{bmatrix}
</math>

---

== Important Notes ==
* You cannot add matrices of different dimensions.
* Matrix addition is commutative:
<math>
A + B = B + A
</math>
* It is also associative:
<math>
(A + B) + C = A + (B + C)
</math>

---

''Matrix addition is a simple but essential operation in linear algebra and is widely used in computer science, engineering, and physics.''