Q.If possible, find the sum of the matrices $A$ and $B$, where $A = \begin{bmatrix} \sqrt{3} & 1 \\ 2 & 3 \end{bmatrix}$, and $B = \begin{bmatrix} x & y & z \\ a & b & 6 \end{bmatrix}$.

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

Step 1: Identify the orders.

$A$ is $2 \times 2$ (2 rows, 2 columns).

$B$ is $2 \times 3$ (2 rows, 3 columns).

Step 2: Since the number of columns differs (2 vs. 3), the matrices are not conformable for addition. …

Matrix addition is one of the simplest operations in linear algebra, but it has a strict rule: you can only add matrices that are exactly the same size. Think of it like adding two tables of numbers — each position in the first table must have a matching position in the second. If the tables have different shapes, there’s no sensible way to pair up the entries.

Let’s check the orders of the given matrices.

1. Identify the order of $A$

   $A = \begin{bmatrix} \sqrt{3} & 1 \\ 2 & 3 \end{bmatrix}$ has 2 rows and 2 columns. So its order is $2 \times 2$.

2. Identify the order of $B$

   $B = \begin{bmatrix} x & y & z \\ a & b & 6 \end{bmatrix}$ has 2 rows and 3 columns. So its order is $2 \times 3$.

3. Compare the orders

   For addition $A + B$ to be possible, both matrices must have the same number of rows and the same number of columns. Here, $A$ has 2 columns but $B$ has 3 columns. They do not match. …

Method: Testing whether a matrix sum is defined

Use this before adding any two matrices — the order check comes first.

Steps

Step 1: Write both orders.

Read off $m_1\times n_1$ and $m_2\times n_2$.

Step 2: Compare orders exactly.

Addition is defined only when the two orders are identical:

Step 3: Conclude honestly. …

Common Mistakes

Mistake 1: Adding matrices of different orders anyway.

Why it's wrong: $A$ is $2\times2$ and $B$ is $2\times3$, so there is no entry in $A$ to pair with $B$'s third column. Correct approach: declare the sum undefined.

Mistake 2: Padding the smaller matrix with zeros to force a sum. …