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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]|`

Text Solution

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`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]|`
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