Q.If $A$ and $B$ are two square matrices of the same order, then $A + B = B + A$.

Concept: Matrix Addition — addition of matrices is defined element-wise, and since real numbers commute under addition, the same holds for matrices.

Step 1: Let $A = [a_{ij}]$ and $B = [b_{ij}]$ be two $n \times n$ matrices.

Step 2: By definition, $(A + B)_{ij} = a_{ij} + b_{ij}$. …

The statement in the question is a fundamental property of matrix addition. Let's understand why it's true, not just memorize it.

The Core Idea: Entry-by-Entry Addition

When you add two matrices, you're really just adding numbers in corresponding positions. Think of it like this: if you have two spreadsheets with the same layout, adding them means adding the number in cell (1,1) of the first to the number in cell (1,1) of the second, and so on for every cell.

Matrix addition is defined entry-wise. For two matrices $A$ and $B$ of the same order (say $m \times n$), their sum $A+B$ is another $m \times n$ matrix where each entry is:

Here $A_{ij}$ means the entry in the $i$-th row and $j$-th column of $A$, and similarly for $B$.

Why Commutativity Follows Naturally

Now, the commutativity of matrix addition — that $A+B = B+A$ — comes directly from the commutativity of ordinary addition of numbers. Let's walk through it:

1. Take any position $(i,j)$ in the matrices. In $A+B$, the entry here is $A_{ij} + B_{ij}$.

2. In $B+A$, the entry at the same position is $B_{ij} + A_{ij}$.

3. But for ordinary numbers, we know $A_{ij} + B_{ij} = B_{ij} + A_{ij}$. This is the commutative property of real (or complex) numbers — it's something we use without thinking when we say $5+3 = 3+5$.

4. Since this holds for every single position $(i,j)$ in the matrices, every corresponding entry in $A+B$ and $B+A$ is equal. Two matrices are equal precisely when all their corresponding entries match.

5. Therefore, $A+B = B+A$ as matrices.

A Quick Example to See It

Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$.

Then $A+B = \begin{pmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}$. …

Method: Proving a Matrix Property by Reducing to Corresponding Entries

The cleanest way to prove an addition/subtraction property of matrices is to look at one arbitrary entry and fall back on ordinary-number algebra, since these operations are defined entrywise.

Steps

Step 1: Confirm the operation is entrywise and the orders match.

Addition is only defined for matrices of the same order, and

Step 2: Prove the property at the level of a single entry $(i,j)$.

Whatever the claimed identity, evaluate both sides at position $(i,j)$. Here $(A+B)_{ij}=a_{ij}+b_{ij}$ and $(B+A)_{ij}=b_{ij}+a_{ij}$.

Step 3: Invoke the matching property of numbers, then generalise. …

Common Mistakes

Mistake 1: Carrying the commutativity of addition over to multiplication.

Why it's wrong: $A+B=B+A$ is always true, but $AB=BA$ is not — the two operations behave differently. Correct approach: prove addition-commutativity entrywise, and keep multiplication separate.

Mistake 2: Forgetting the same-order requirement. …