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  • Construct a 3 × 2 matrix whose elements are given by a_ij = e^ixsin jx

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

 

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A matrix, in mathematics is a rectangular array of numbers, alphabets, symbols, or expressions, arranged in rows and columns.

Also,

We know that, the notation A = [a\textsubscript{ij}]\textsubscript{m$ \times $ m} indicates that the matrix A has the order of A m \times n, also 1 \leq i \leq m, 1 \leq j \leq n; i, j \in \mathbb{N}.

We need to construct a 3 \times 2  matrix whose elements are as follows:

a\textsubscript{ij} = e\textsuperscript{i.x} sin jx

For a\textsubscript{3$ \times $ 2}:

\\ 1 \leq i \leq m \\ \Rightarrow 1 \leq i \leq 3 \ [\because m = 3] \\ 1 \leq j \leq n \\ \Rightarrow 1 \leq j \leq 2 \ [\because n = 2]

Put i = 1 and j = 1.

\\a\textsubscript{11} = e\textsuperscript{(1)x} sin (1)x \\ \Rightarrow a\textsubscript{11} = e\textsuperscript{x} sin x

Put i = 1 and j = 2.

 

\\a\textsubscript{12} = e\textsuperscript{(1)x} sin (2)x \\ \Rightarrow a\textsubscript{12} = e\textsuperscript{x} sin 2x

 

Put i = 2 and j = 1.

 

\\a\textsubscript{21} = e\textsuperscript{(2)x} sin (1)x \\ \Rightarrow a\textsubscript{21} = e\textsuperscript{2x}sin x

 

Put i = 2 and j = 2.

 \\a\textsubscript{22} = e\textsuperscript{(2)x} sin (2)x \\ \Rightarrow a\textsubscript{22} = e\textsuperscript{2x} sin 2x

For i = 3 and j = 1.

 \\a\textsubscript{31} = e\textsuperscript{(3)x} sin (1)x \\ \Rightarrow a\textsubscript{31} = e\textsuperscript{3x} sin x

For i = 3 and j = 2.

\\a\textsubscript{32} = e\textsuperscript{(3)x} sin (2)x \\ \Rightarrow a\textsubscript{32} = e\textsuperscript{3x} sin 2x

Let the matrix formed be A.

\begin{aligned} &A=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{array}\right]\\ &\text { By substituting the values of } a_{11}, a_{12}, a_{21}, a_{22}, a_{31} \text { and } a_{32}, \text { we get the following matrix }\\ &A=\left[\begin{array}{ll} e^{x} \sin x & e^{x} \sin 2 x \\ e^{2 x} \sin x & e^{2 x} \sin 2 x \\ e^{3 x} \sin x & e^{3 x} \sin 2 x \end{array}\right] \end{aligned}

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