Q.Matrices of any order can be added.

Concept: Matrix Addition — two matrices can be added only if they have the same order (same number of rows and columns). Addition is performed element-wise.

Reasoning:

1. The statement says "Matrices of any order can be added."
2. For matrix addition to be defined, both matrices must have identical dimensions. For example, a $2 \times 3$ matrix cannot be added to a $3 \times 2$ matrix. …

The Core Idea: Why Order Matters

Matrix addition is not like adding numbers. When you add two matrices, you are combining information that is structured in a specific grid. For the addition to make sense, the two grids must be identical in shape — same number of rows, same number of columns. This is because we add entry by entry: the element in row $i$, column $j$ of the first matrix gets added to the element in row $i$, column $j$ of the second matrix.

If the matrices have different orders, there will be positions in one matrix that have no corresponding position in the other. You cannot add something to nothing.

Step-by-Step Reasoning

1. Definition of matrix addition.

   Let $A = [a_{ij}]$ be an $m \times n$ matrix and $B = [b_{ij}]$ be a $p \times q$ matrix. The sum $A + B$ is defined only if $m = p$ and $n = q$ — that is, both matrices have the same number of rows and the same number of columns. When this condition holds, the sum $C = A + B$ is an $m \times n$ matrix where each entry $c_{ij} = a_{ij} + b_{ij}$.

2. Why "any order" fails.

   Consider a $2 \times 3$ matrix and a $2 \times 2$ matrix. The first has 6 entries, the second has 4 entries. There is no way to pair up all entries for addition. For example, the element in row 1, column 3 of the first matrix has no counterpart in the second matrix. The operation is simply not defined.

3. A concrete counterexample.

   Let $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ (order $2 \times 3$) and $B = \begin{pmatrix} 7 & 8 \\ 9 & 10 \end{pmatrix}$ (order $2 \times 2$).

   Can we add $A$ and $B$? No — because $A$ has 3 columns while $B$ has only 2. There is no entry in $B$ to add to $a_{13} = 3$ or $a_{23} = 6$.

4. The only exception is trivial. …

Method: Check whether matrix addition is defined

Use this to judge any claim about when two matrices can be added.

Steps

Step 1: State the rule for addition.

Addition is defined only for matrices of the same order (same number of rows and same number of columns), and is carried out entry-by-entry.

Step 2: Test the claim against the rule.

For a "can any matrices be added?" statement, look for a pair of different orders — if such a pair fails, the universal claim is false. …

Common Mistakes

Mistake 1: Thinking equal total number of entries is enough to add.

Why it's wrong: a $2\times3$ and a $3\times2$ both have $6$ entries but the positions do not correspond, so addition is undefined. Correct approach: require the same order, not just the same count of entries.

Mistake 2: Ignoring positional correspondence. …