Question

# The sum of three numbers is 6. If we multiply the third number by 3 and add it to the second number to it, we get 11. By adding first and third numbers, we get double of the second number. Represent it algebraically and find the numbers using matrix method.

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Solution

## Let the first , second and third number be x,y,z respectively. Then, according to the given condition, we have x+y+z=6y+3z=11 x+z=2y or x−2y+z=0 This system of equations can be written as AX=B, where A=⎡⎢⎣1110131−21⎤⎥⎦,X=⎡⎢⎣xyz⎤⎥⎦& B=⎡⎢⎣6110⎤⎥⎦ A=1(1+6)−0+1(3−1) =9 ⇒|A|≠0 ∴ The system of equation is consistent and has a unique solution. Now, we find adjA A11=7, A12=3, A13=−1 A21=−3, A22=0, A23=3 A31=2, A32=−3, A33=1 Hence , adj(A)=⎡⎢⎣7−3230−3−131⎤⎥⎦ Thus A−1=1|A|adj(A) =19⎡⎢⎣7−3230−3−131⎤⎥⎦ Since, AX=B ∴X=A−1B ⇒X=19⎡⎢⎣7−3230−3−131⎤⎥⎦⎡⎢⎣6110⎤⎥⎦ ⇒⎡⎢⎣xyz⎤⎥⎦=19⎡⎢⎣91827⎤⎥⎦=⎡⎢⎣123⎤⎥⎦ ⇒x=1,y=2,z=3

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