Q.Construct a $3 \times 2$ matrix whose elements are given by $a_{ij} = e^{ix}\sin jx$.

Concept: Matrix Construction — each entry is defined by a formula depending on row index $i$ and column index $j$.

Step 1: The matrix is $3 \times 2$, so $i = 1,2,3$ and $j = 1,2$.

Step 2: The element $a_{ij} = e^{ix} \sin(jx)$. For each row $i$, the factor $e^{ix}$ is constant across columns; only $\sin(jx)$ changes with $j$.

Step 3: Compute each entry:

• Row $i=1$: $a_{11} = e^{x}\sin x$, $a_{12} = e^{x}\sin 2x$
• Row $i=2$: $a_{21} = e^{2x}\sin x$, $a_{22} = e^{2x}\sin 2x$ …

When you see a problem like "construct a matrix whose elements are given by $a_{ij} = \text{something}$", the first thing to realise is that you are not solving an equation — you are following a recipe. The indices $i$ and $j$ are just placeholders for the row number and column number. So the matrix is built by plugging in $i = 1, 2, 3$ and $j = 1, 2$ into the expression $e^{ix}\sin jx$.

The expression $e^{ix}\sin jx$ is a product of two factors: one depends only on $i$ (the row), the other depends only on $j$ (the column). This is a special kind of matrix called a separable or rank-1 matrix — but you don't need that term for the exam. What matters is that you can fill each entry independently.

Let's do it step by step.

1. Identify the dimensions. The matrix is $3 \times 2$, meaning 3 rows and 2 columns. So $i$ takes values $1, 2, 3$ and $j$ takes values $1, 2$.

2. Write the general entry. For any row $i$ and column $j$, the entry is $a_{ij} = e^{ix}\sin jx$. Notice that $x$ is a variable (presumably a real number or a parameter), so the matrix entries are functions of $x$.

3. Fill the first column ($j = 1$).

   • Row 1 ($i=1$): $a_{11} = e^{1 \cdot x}\sin(1 \cdot x) = e^{x}\sin x$
   • Row 2 ($i=2$): $a_{21} = e^{2x}\sin x$
   • Row 3 ($i=3$): $a_{31} = e^{3x}\sin x$

4. Fill the second column ($j = 2$).

   • Row 1 ($i=1$): $a_{12} = e^{x}\sin(2x)$
   • Row 2 ($i=2$): $a_{22} = e^{2x}\sin 2x$
   • Row 3 ($i=3$): $a_{32} = e^{3x}\sin 2x$

5. Assemble the matrix. Place each entry in its correct position:

$\begin{pmatrix} e^{x}\sin x & e^{x}\sin 2x \\[4pt] e^{2x}\sin x & e^{2x}\sin 2x \\[4pt] e^{3x}\sin x & e^{3x}\sin 2x \end{pmatrix}$ …

Method: Constructing a matrix from a function-valued formula

Use this when the entry rule $a_{ij}=f(i,j)$ contains functions (here $e^{ix}\sin jx$) rather than plain arithmetic.

Steps

Step 1: Fix the shape from the order.

A $3\times2$ matrix has $i\in\{1,2,3\}$ (rows) and $j\in\{1,2\}$ (columns). Build a $3\times2$ grid, not $2\times3$.

Step 2: Read which index sits where in the formula. …

Common Mistakes

Mistake 1: Building the wrong shape.

Why it's wrong: a $3\times2$ matrix has 3 rows and 2 columns ($i=1,2,3$, $j=1,2$); producing a $2\times3$ grid is a different matrix. Correct approach: let the order set the index ranges before substituting.

Mistake 2: Misreading the indices inside the functions. …