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: $A=[a_{ij}],B=[b_{ij}]$

Then: $A+B=[a_{ij}+b_{ij}]$

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

Step-by-Step Example

Let:

$A=\left[\begin{array}{ccc}2& 4& 6\\ 1& 3& 5\end{array}\right],B=\left[\begin{array}{ccc}7& 1& 3\\ 4& 0& 2\end{array}\right]$

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

Step 2: Add corresponding elements:

$A+B=\left[\begin{array}{ccc}2+7& 4+1& 6+3\\ 1+4& 3+0& 5+2\end{array}\right]$

Step 3: Perform the additions:

$A+B=\left[\begin{array}{ccc}9& 5& 9\\ 5& 3& 7\end{array}\right]$

Final Answer

$\left[\begin{array}{ccc}9& 5& 9\\ 5& 3& 7\end{array}\right]$

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Important Notes

• You cannot add matrices of different dimensions.
• Matrix addition is commutative:

$A+B=B+A$

• It is also associative:

$(A+B)+C=A+(B+C)$

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