Taking out a, b and c from the determinant from 1st, 2nd and 3rd row respectively.


R2 R2 - R1 (i.e. replace 2nd row by subtraction of 2nd and 1st row)


R3 R3 - R1 (i.e. replace 3rd row by subtraction of 3rd and 1st row)


.


C1 aC1 (i.e. replace 1st column by ‘a’ times 1st column)


C2 bC2 (i.e. replace 2nd column by ‘b’ times 2nd column)


cC3 (i.e. replace 3rd column by ‘c’ times 3rd column)


As we are multiplying by a, b and c we should also divide by a, b and c to keep the original determinant value unchanged.


.


C1 C1 + C2 + C3 (i.e. replace 1st column by addition of 1st, 2nd and 3rd column)



Expanding determinant along 1st column



LHS = (1 + a2 + b2 + c2) = RHS



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