Q.In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17 \end{bmatrix}$, write:

Concept: Matrix Notation — A matrix is described by its order (rows × columns), and each element $a_{ij}$ is located at row $i$, column $j$.

Step 1: Order of the matrix

Count rows: 3. Count columns: 4.

So the order is $3 \times 4$.

Step 2: Number of elements

Multiply rows by columns: $3 \times 4 = 12$.

Step 3: Identify the required elements

• $a_{13}$: row 1, column 3 → $19$
• $a_{21}$: row 2, column 1 → $35$
• $a_{33}$: row 3, column 3 → $-5$
• $a_{24}$: row 2, column 4 → $12$
• $a_{23}$: row 2, column 3 → $\frac{5}{2}$

The first thing to understand is what the notation $a_{ij}$ means. In any matrix, $a_{ij}$ refers to the element in the $i$-th row and $j$-th column. The row number always comes first, the column number second. This is the universal convention — think of it like coordinates on a grid, where you go down first, then across.

The matrix given is:

Let's work through each part.

1. Order of the matrix

   Count the rows: there are 3 horizontal lines of entries. Count the columns: each row has 4 entries. So the order is $3 \times 4$ (read as "3 by 4"). The order is always written as (number of rows) $\times$ (number of columns).

2. Number of elements

   The total number of entries is simply rows $\times$ columns. That gives $3 \times 4 = 12$ elements. You can verify by counting: each row has 4 numbers, and there are 3 rows.

3. Finding specific elements

   Now we locate each requested entry by its row and column indices.

   • $a_{13}$: row 1, column 3. In the first row, the third entry is $19$.
   • $a_{21}$: row 2, column 1. The first entry of the second row is $35$.
   • $a_{33}$: row 3, column 3. The third entry of the third row is $-5$.
   • $a_{24}$: row 2, column 4. The fourth entry of the second row is $12$.
   • $a_{23}$: row 2, column 3. The third entry of the second row is $\frac{5}{2}$.

Method: Reading order, element count, and specific entries

Use this for any "state the order / number of elements / value of $a_{ij}$" question.

Steps

Step 1: Find the order.

Count the horizontal rows ($m$) and the vertical columns ($n$). The order is written rows-first: $m\times n$.

Step 2: Find the number of elements.

It is simply the product $m\times n$.

Step 3: Locate each entry by its indices.

The symbol $a_{ij}$ means row $i$ first, then column $j$. Move down to row $i$, then across to column $j$, and read the value there — even if the entry is a fraction, surd, or variable expression.

Common Mistakes

Mistake 1: Swapping the row and column indices.

Why it's wrong: $a_{21}$ is row 2, column 1 $(=35)$, not row 1, column 2 $(=5)$. Correct approach: read $a_{ij}$ as row $i$ first, then column $j$.

Mistake 2: Writing the order as columns $\times$ rows.

Why it's wrong: order is always rows $\times$ columns, so this matrix is $3\times4$, not $4\times3$. Correct approach: count rows first.

Mistake 3: Miscounting elements.

Why it's wrong: the count is $m\times n=12$, not $m+n$. Correct approach: multiply the order's two numbers.