the amount of Rs.10000/- is put into three investments at the rate of 10% ,12% and 15% per annum ,the combined income is Rs. 1290 and the combined income from the first and second is Rs. 210 less than the third . find the investment in each using matrix method

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Let x,y and z be three investments at the rate of 10% ,12% and 15% per annum respectively.x+y+z=10000    ...(1)The income on sum Rsx isx×10×1100=10x100The combined income is10x100+12y100+15z100=1290The equation can be re-written as:10x+12y+15z=129000    ...(2)Since the combined income of 1st and 2nd is Rs210 less than the income on 3rd sum.10x100+12y100=15z100-21010x+12y-15z=-21000    ...(3)The equation (1),(2) and (3) can be written in the form of AX=B1111012151012-15xyz=10000129000-21000where A=1111012151012-15,B=10000129000-21000 and X=xyzNow to check whether the solution is consistent, we will find determinant of A.A=1111012151012-15=1001025102-25=-50-10=-60   By using C2C2-C1 and C3C3-C1Since dtA is not equal to zero, therefore solution exist.We will solve this system of equation by Cramer's Rule.D=-60D1=10000111290001215-2100012-15=100010111291215-2112-15=10000109123-14112-37    C1C1-10C2 and  C3C3-C2=1000-423+333=-90000D2=1100001101290001510-21000-15=10001101101291510-21-15=1000001-5-211525129-15           C1C1-C3 and  C2C2-10C3=1000-645+525=-120000D3=111000010121290001012-21000=1000111010121291012-21=100010010229102-121       C2C2-C1 and  C3C3-10C1=1000-242-58=-300000Therefore,x= D1D=-90000-60=1500y= D2D=-120000-60=2000z= D3D=-300000-60=5000Thus, the investment at 10%,12% and 15% per annum are Rs1500,Rs2000 and Rs5000

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