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