Q.Matrix addition is associative as well as commutative.

Concept: Matrix Addition — addition is performed element-wise, and the properties follow directly from the properties of real number addition.

Reasoning:

1. For two $m \times n$ matrices $A$ and $B$, addition is defined as $(A + B)_{ij} = a_{ij} + b_{ij}$.
2. Commutativity: Since $a_{ij} + b_{ij} = b_{ij} + a_{ij}$ for every entry (real numbers commute), we have $A + B = B + A$. …

Why This Works: The Core Idea

Matrix addition is the simplest operation in linear algebra — and that simplicity is exactly why these properties hold. When you add two matrices, you add their corresponding entries one by one. There is no cross-talk between entries, no multiplication, no fancy rules. Each entry behaves independently.

So the question of whether matrix addition is associative or commutative reduces to a much simpler question: Is the addition of ordinary numbers associative and commutative? The answer is yes — and that's the entire proof.

Let's see this in detail.

Step-by-Step Reasoning

1. Define what we mean by matrix addition.

If $A$ and $B$ are both $m \times n$ matrices, their sum $A+B$ is another $m \times n$ matrix where each entry is the sum of the corresponding entries:

for every row $i$ and column $j$. This definition is the foundation of everything that follows.

2. Check commutativity: $A+B = B+A$.

Take any entry at position $(i,j)$ in the sum $A+B$. By definition, it is $A_{ij} + B_{ij}$. Now consider the same entry in $B+A$: it is $B_{ij} + A_{ij}$.

Since addition of real numbers is commutative, $A_{ij} + B_{ij} = B_{ij} + A_{ij}$ for every $i$ and $j$. Therefore every entry of $A+B$ equals the corresponding entry of $B+A$, so the matrices are identical:

3. Check associativity: $(A+B)+C = A+(B+C)$.

Now let $A$, $B$, $C$ all be $m \times n$ matrices. Consider the entry at position $(i,j)$ in $(A+B)+C$.

First, $(A+B)_{ij} = A_{ij} + B_{ij}$. Then adding $C$ gives:

Now look at the same entry in $A+(B+C)$. First, $(B+C)_{ij} = B_{ij} + C_{ij}$. Then adding $A$ gives:

Since addition of real numbers is associative, $(A_{ij} + B_{ij}) + C_{ij} = A_{ij} + (B_{ij} + C_{ij})$ for every $i$ and $j$. Hence every entry matches, and the matrices are equal:

4. What about the zero matrix?

The $m \times n$ zero matrix, denoted $0$, has all entries equal to $0$. It acts as the additive identity: …

Method: Prove an entrywise operation inherits scalar properties

Use this to justify why matrix addition is commutative and associative.

Steps

Step 1: Use the entrywise definition.

For same-order matrices, $(A+B)_{ij} = a_{ij} + b_{ij}$ — addition happens independently at each position.

Step 2: Push the property down to a single entry.

Because real-number addition is commutative and associative, at each position …

Common Mistakes

Mistake 1: Assuming addition might fail to commute because matrix multiplication does not.

Why it's wrong: addition and multiplication are different operations; addition is entrywise and inherits commutativity/associativity from real numbers. Correct approach: judge each operation by its own definition. …