Matrix Addition

Matrix Addition – Step-by-Step Explanation

Matrix addition is a method of combining two matrices by adding the elements that are in the same position. This operation is only defined when both matrices have the same dimensions (same number of rows and columns).

Step 1: Understand Matrix Dimensions

Each matrix has an "order" or "size" defined by the number of rows and columns it has. A matrix with 2 rows and 3 columns is a 2×3 matrix.

Example:

Let’s take two matrices A and B:

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

Matrix A is a 2×3 matrix (2 rows, 3 columns) Matrix B is also a 2×3 matrix → ✅ Same size ⇒ Matrix addition is possible

Step 2: Identify Corresponding Elements

Matrix addition is performed by adding corresponding elements in each position.

So we pair each element in matrix A with the one in matrix B at the same location:

• (1,1) → $2+7$
• (1,2) → $4+1$
• (1,3) → $6+3$
• (2,1) → $1+4$
• (2,2) → $3+0$
• (2,3) → $5+2$

Step 3: Perform the Addition

Now add each pair:

$\begin{array}{rl}2+7& =9\\ 4+1& =5\\ 6+3& =9\\ 1+4& =5\\ 3+0& =3\\ 5+2& =7\end{array}$

Step 4: Write the Resulting Matrix

Now place the results in their original positions to form the new matrix:

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

✅ Final Answer

Failed to parse (syntax error): {\displaystyle \boxed{ \begin{bmatrix} 9 & 5 & 9 \\ 5 & 3 & 7 \end{bmatrix} } }

Important Properties of Matrix Addition

• Commutative Law: $A+B=B+A$
• Associative Law: $(A+B)+C=A+(B+C)$
• The zero matrix $O$ acts like 0:

 $A+O=A$

Summary

To add matrices: 1. Confirm that both matrices are the same size. 2. Add elements in corresponding positions. 3. Write the results in a new matrix of the same size.

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Matrix addition is a basic yet powerful tool used in algebra, data science, and applied mathematics!