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प्रश्न
Four men and 4 women can finish a piece of work jointly in 3 days while 2 men and 5 women can finish the same work jointly in 4 days. Find the time taken by one man alone and that of one woman alone to finish the same work by using matrix inversion method
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उत्तर
Let the time taken by one man alone be x days and one woman alone be y days.
4x + 4y = `1/3`
2x + 5y = `1/4`
Matrix form `[(4, 4),(2, 5)] [(x),(y)] = [(1/3),(1/4)]`
AX = B
X = `"A"^-1 "B"`
A = `[(4, 4),(2, 5)]`
|A|+ 20 – 8
= 12 ≠ 0.A–1 exists.
adj A = `[(5, -4),(-2, 4)]`
A–1 = `1/|"A"|`
adj A = `1/12[(5, -4),(-2, 4)]`
X = `1/12[(5, -4),(-2, 4)][(1/3),(1/4)]`
= `1/12[(5/3 - 1),((-2)/3 + 1)]`
= `1/12 [(2/3),(1/3)]`
= `[(1/18),(1/36)]`
`[(x),(y)] = [(1/18),(1/36)]`
∴ One man can do 18 days
One woman can do 36 days
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