Question

Read the following passage:

Gautam buys 5 pens, 3 bags and 1 instrument box and pays a sum of ₹160. From the same shop, Vikram buys 2 pens, 1 bag and 3 instrument boxes and pays a sum of ₹190. Also, Ankur buys 1 pen, 2 bags and 4 instrument boxes and pays a sum of ₹250.

Based on the above information, answer the following questions:

1. Convert the given above situation into a matrix equation of the form AX = B. (1)
2. Find | A |. (1)
3. Find A^–1. (2)
   OR
   Determine P = A² – 5A. (2)

Answer 1

5x + 3y + z = 160

2x + y + 3z = 190

x + 2y + 4z = 250

I. AX = B form

`[(5, 3, 1),(2, 1, 3),(1, 2, 4)][(x),(y),(z)] = [(160),(190),(250)]`

II. | A | = `|(5, 3, 1),(2, 1, 3),(1, 2, 4)|`

= 5(4 – 6) – 3(8 – 3) + 1(4 – 1)

= 5(– 2) – 3(5) + 3

= – 10 – 15 + 3

= – 22.

III. A^–1 = `1/|A|` (adj A)

 C₁₁ = – 2, C₁₂ = – 5, C₁₃ = 3

C₂₁ = – 10, C₂₂ = 19, C₂₃ = – 7

C₃₁ = 8, C₃₂ = – 13, C₃₃ = – 1

Adj A = `[(-2, -10, 8),(-5, 19, -13),(3, -7, -1)]`

A^–1 = `1/|A|` (adj A)

= `1/-22 [(-2, -10, 8),(-5, 19, -13),(3, -7, -1)]`

A^–1 = `1/22 [(2, 10, -8),(5, -19, 13),(-3, 7, 1)]`

OR

P = A² – 5A

= `[(5, 3, 1),(2, 1, 3),(1, 2, 4)][(5, 3, 1),(2, 1, 3),(1, 2, 4)] - 5[(5, 3, 1),(2, 1, 3),(1, 2, 4)]`

= `[(32, 20, 18),(15, 13, 17),(13, 13, 23)] - [(25, 15, 5),(10, 5, 15),(5, 10, 20)]`

= `[(7, 5, 13),(5, 8, 2),(8, 3, 3)]`