Construct a 3 4 matrix, whose elements are given by aij = 2i - j

Question

Construct a 3 $\times$ 4 matrix, whose elements are given by a_ij = 2i - j

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Answer

In general 3 $\times$ 4 matrix is given by A = $\left[\begin{array}{llll} {a_{11}} & {a_{12}} & {a_{13}} & {a_{14}} \\ {a_{21}} & {a_{22}} & {a_{23}} & {a_{24}} \\ {a_{31}} & {a_{32}} & {a_{33}} & {a_{34}} \end{array}\right]$
a_ij = 2i - j, i = 1, 2, 3 and j = 1, 2, 3, 4
Therefore,
a₁₁ = 2 $\times$ 1 - 1 = 2 - 1 = 1
a₂₁ = 2 $\times$ 2 - 1 = 4 - 1 = 3
a₃₁ = 2 $\times$ 3 - 1 = 6 - 1 = 5
a₁₂ = 2 $\times$ 1 - 2 = 2 - 2 = 0
a₂₂ = 2 $\times$ 2 - 2 = 4 - 2 = 2
a₃₂ = 2 $\times$ 3 - 2 = 6 - 2 = 4
a₁₃ = 2 $\times$ 1 - 3 = 2 - 3 = -1
a₂₃ = 2 $\times$ 2 - 3 = 4 - 3 = 1
a₃₃ = 2 $\times$ 3 - 3 = 6 - 3 = 3
a₁₄ = 2 $\times$ 1 - 4 = 2 - 4 = -2
a₂₄ = 2 $\times$ 2 - 4 = 4 - 4 = 0
a₃₄ = 2 $\times$ 3 - 4 = 6 - 4 = 2
Therefore, required matrix is A = $\left[\begin{array}{cccc} {1} & {0} & {-1} & {-2} \\ {3} & {2} & {1} & {0} \\ {5} & {4} & {3} & {2} \end{array}\right]$

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