Q.Consider the following information regarding the number of men and women workers in three factories I, II and III. Men workers / Women workers: Factory I: $30$, $25$; Factory II: $25$, $31$; Factory III: $27$, $26$. Represent the above information in the form of a $3 \times 2$ matrix. What does the entry in the third row and second column represent?

Concept: Matrix Notation — arranging data in rows and columns where each entry has a specific meaning based on its position.

Step 1: Identify the rows and columns. Each row corresponds to a factory (I, II, III), and each column corresponds to a worker category (first column = men, second column = women).

Step 2: Write the data row-wise:

Factory I: (30, 25)

Factory II: (25, 31)

Factory III: (27, 26)

Step 3: Assemble into a $3 \times 2$ matrix:

The entry in the third row and second column is $26$, which is the number of women workers in Factory III.

Why a matrix? The idea of structured data

When you have information that naturally falls into rows and columns — like factories (rows) and worker categories (columns) — a matrix is the cleanest way to store and read it. Each entry $(i, j)$ tells you the value at the intersection of the $i$-th row and $j$-th column.

Here, we have three factories (I, II, III) and two categories (Men, Women). That’s a $3 \times 2$ matrix: 3 rows, 2 columns.

Step-by-step construction

1. Identify the row labels and column labels.

Rows: Factory I, Factory II, Factory III.

Columns: Column 1 = Men, Column 2 = Women.

2. Fill the first row (Factory I).

Men = 30, Women = 25.

So row 1 is $[30 \quad 25]$.

3. Fill the second row (Factory II).

Men = 25, Women = 31.

Row 2 is $[25 \quad 31]$.

4. Fill the third row (Factory III).

Men = 27, Women = 26.

Row 3 is $[27 \quad 26]$.

5. Write the complete matrix.

Putting it all together:

This is the required $3 \times 2$ matrix.

Interpreting the entry in the third row, second column

The third row corresponds to Factory III.

The second column corresponds to Women workers.

So the entry at row 3, column 2 is $26$, which means: the number of women workers in Factory III is 26.

Method: Representing tabular data as a matrix and reading an entry $a_{ij}$

Use this for problems that give a table (rows = categories, columns = attributes) and ask for the matrix plus the meaning of a specific position.

Steps

Step 1: Fix what rows and columns represent.

Decide the labelling before writing anything — e.g. rows = factories, columns = worker type. The required order (say $3\times2$) tells you rows $\times$ columns.

Step 2: Fill the grid row by row.

Place each data value at its correct (row, column) position, keeping the stated order.

Step 3: Interpret $a_{ij}$ by its address.

The entry in row $i$, column $j$ is $a_{ij}$; state what that row and that column stand for (e.g. $a_{32}$ = women workers in Factory III).

Common Mistakes

Mistake 1: Writing the transpose shape (e.g. $2\times3$ instead of $3\times2$).

Why it's wrong: the order is fixed as rows $\times$ columns; swapping them changes every entry's address. Correct approach: match the requested order exactly — 3 rows, 2 columns here.

Mistake 2: Confusing $a_{32}$ with $a_{23}$.

Why it's wrong: the first index is always the row, the second the column; reversing them points to the wrong data. Correct approach: read $a_{ij}$ as row $i$, column $j$.

Mistake 3: Interpreting the entry as just a number, not its meaning.

Why it's wrong: the question asks what the entry represents, not only its value. Correct approach: name the row-category and column-attribute together (women workers, Factory III).