Given a,b,c integer show that the system have unique solution and find solution? (a+(1/3))x+by+cz=0 ax+(b+(1/3))y+cz=0 ax+by+(c+(1/3))z=0

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llltkl | College Teacher | (Level 3) Valedictorian

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Given a,b,c are integers, to show that the system have unique solution we have to find the value of the following determinant:


The value of this determinant is:


`=3abc+(ab)/3+(bc)/3+(ac)/3+a/9+b/9+c/9+1/27-3abc -(ab)/3-(bc)/3-(ac)/3`



Since, this determinant is non-zero, by Cramer's rule the given system of equations have a unique solution.

Hence, x=0, y=0 and z=0 since each of the determinants `D_x` , `D_y` ,`D_z ` will contain a column of zeros (i.e the constant terms of the three equations).


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