InterviewSolution
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InterviewSolution
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Using the property of determinants and without expanding, prove that:`|[b+c, q+r, y+z],[ c+a, r+p, z+x],[ a+b, p+q, x+y]|=2|[a, p, x],[ b, q ,y],[ c, r, z]|` |
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Answer» `L.H.S. = |[b+c,q+r,y+z],[c+A,r+p,z+x],[a+b,p+q,x+y]|` Applying `R_1->R_1+R_2+R_3` `= |[2(a+b+c),2(p+q+r),2(x+y+z)],[c+a,r+p,z+x],[a+b,p+q,x+y]|` `= 2|[a+b+c,p+q+r,x+y+z],[c+a,r+p,z+x],[a+b,p+q,x+y]|` Applying `R_2->R_2-R_1` and `R_3->R_3-R_1` `=2|[a+b+c,p+q+r,x+y+z],[-b,-q,-y],[-c,-r,-z]|` Applying `R_1->R_1+R_2+R_3` `=2|[a,p,x],[-b,-q,-y],[-c,-r,-z]|` `=2(-1)(-1)|[a,p,x],[b,q,y],[c,r,z]|` `=2|[a,p,x],[b,q,y],[c,r,z]|=R.H.S.` |
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