A matrix, as we know, is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

Also,

We know that, the notation A = [a_ij]_m×m indicates that A is a matrix of order m × n, also 1 ≤ i ≤ m, 1 ≤ j ≤ n; i, j ∈ N.

We need to construct a 3 × 2 matrix whose elements are given by

a_ij = e^i.x sin jx

For a_3×2:

1 ≤ i ≤ m

⇒ 1 ≤ i ≤ 3 [∵ m = 3]

1 ≤ j ≤ n

⇒ 1 ≤ j ≤ 2 [∵ n = 2]

Put i = 1 and j = 1.

a₁₁ = e^(1)x sin (1)x

⇒ a₁₁ = e^x sin x

Put i = 1 and j = 2.

a₁₂ = e^(1)x sin (2)x

⇒ a₁₂ = e^x sin 2x

Put i = 2 and j = 1.

a₂₁ = e^(2)x sin (1)x

⇒ a₂₁ = e^2xsin x

Put i = 2 and j = 2.

a₂₂ = e^(2)x sin (2)x

⇒ a₂₂ = e^2x sin 2x

For i = 3 and j = 1.

a₃₁ = e^(3)x sin (1)x

⇒ a₃₁ = e^3x sin x

For i = 3 and j = 2.

a₃₂ = e^(3)x sin (2)x

⇒ a₃₂ = e^3x sin 2x

Let the matrix formed be A.

Substituting the values of a₁₁, a_12, a₂₁, a₂₂, a₃₁ and a₃₂, we get the matrix

Thus, we have got the matrix.