Question

Construct a 3 × 4 matrix, whose elements are given by:

a_ij = 2i – j

Answer 1

In general, a 3 × 4 matrix is given by A = `[(a_11, a_12, a_13, a_14), (a_21, a_22,  a_23, a_24), (a_31, a_32, a_33, a_34)]`

⇒ a_ij = 2i – j, i = 1, 2, 3 and j = 1, 2, 3, 4

∴ a₁₁ = 2 × 1 – 1

= 2 – 1 

= 1

a₂₁ = 2 × 2 – 1

= 4 – 1 

= 3

a₃₁ = 2 × 3 – 1

= 6 – 1

= 5

⇒ a₁₂ = 2 × 1 – 2

= 2 – 2

= 0

a₂₂ = 2 × 2 – 2

= 4 – 2

= 2

a₃₂ = 2 × 3 – 2

= 6 – 2

= 4

⇒ a₁₃ = 2 × 1 – 3

= 2 – 3

= –1 

a₂₃ = 2 × 2 – 3

= 4 – 3

= 1

a₃₃ = 2 × 3 – 3 

= 6 – 3

= 3

⇒ a₁₄ = 2 × 1 – 4

= 2 – 4

= –2

a₂₄ = 2 × 2 – 4 

= 4 – 4

= 0

a₃₄ = 2 × 3 – 4 

= 6 – 4

= 2

Therefore, the required matrix is A = `[(1, 0, -1, -2),(3, 2, 1, 0),(5, 4, 3, 2)]`.