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Using properties of determinants, prove that `|(b+c,q+r,y+z),(c+a,r+p,z+x),(c+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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