Q.In the matrix $A = \begin{bmatrix} a & 1 & x \\ 2 & \sqrt{3} & x^2 - y \\ 0 & 5 & -\frac{2}{5} \end{bmatrix}$, write:

Concept: Matrix Notation — each element is identified by its row and column position.

Step 1: Order of the matrix

The matrix has 3 rows and 3 columns, so its order is $3 \times 3$.

Step 2: Number of elements

For a $3 \times 3$ matrix, number of elements = $3 \times 3 = 9$.

Step 3: Identifying the required elements

• $a_{23}$: row 2, column 3 → $x^2 - y$
• $a_{31}$: row 3, column 1 → $0$
• $a_{12}$: row 1, column 2 → $1$

Why Matrix Notation Matters

Every matrix is a rectangular array of numbers arranged in rows and columns. The notation $a_{ij}$ is the standard way to refer to the element in the $i$-th row and $j$-th column. Once you understand this indexing system, questions like this become simple lookup exercises — no calculation needed, just careful reading.

The matrix given is:

Let’s walk through each part.

1. The order of the matrix

The order of a matrix is written as $m \times n$, where $m$ is the number of rows and $n$ is the number of columns.

Count the rows: there are 3 horizontal lines of entries.

Count the columns: there are 3 vertical stacks of entries.

So the order is $3 \times 3$.

2. The number of elements

The total number of elements in a matrix of order $m \times n$ is simply $m \times n$.

Here, $m = 3$ and $n = 3$, so:

You can also verify by counting the entries in the matrix: there are 9 numbers (including the variables $a$, $x$, $y$).

3. Writing the specific elements $a_{23}$, $a_{31}$, $a_{12}$

The notation $a_{ij}$ means: go to row $i$, column $j$, and read the entry there.

• $a_{23}$: row 2, column 3.

  Row 2 is: $[2 \quad \sqrt{3} \quad x^2 - y]$.

  Column 3 of that row is $x^2 - y$.

  So $a_{23} = x^2 - y$.

• $a_{31}$: row 3, column 1.

  Row 3 is: $[0 \quad 5 \quad -\frac{2}{5}]$.

  Column 1 of that row is $0$.

  So $a_{31} = 0$.

• $a_{12}$: row 1, column 2.

  Row 1 is: $[a \quad 1 \quad x]$.

  Column 2 of that row is $1$.

  So $a_{12} = 1$.

Method: Reading order, element count, and entries by index

Use this for any question that asks for a matrix's order, its number of elements, and particular entries $a_{ij}$.

Steps

Step 1: Determine the order.

Count rows ($m$) and columns ($n$); the order is $m\times n$, rows first.

Step 2: Count the elements.

The total is $m\times n$, regardless of whether entries are numbers, surds, or algebraic expressions like $x^2-y$.

Step 3: Pick out each requested entry.

For $a_{ij}$, go to row $i$ then column $j$ and copy the entire entry sitting there. Keep the row index first and the column index second.

Common Mistakes

Mistake 1: Reversing the indices.

Why it's wrong: $a_{31}$ means row 3, column 1 $(=0)$, whereas $a_{13}$ would be row 1, column 3 $(=x)$. Correct approach: row index first, column index second.

Mistake 2: Copying only part of an algebraic entry.

Why it's wrong: $a_{23}$ is the whole expression $x^2-y$, not just $x^2$ or $-y$. Correct approach: record the complete entry sitting at that position.

Mistake 3: Confusing "order" with "number of elements".

Why it's wrong: the order is $3\times3$ but the number of elements is $9$. Correct approach: report order as a pair and element-count as their product.