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

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

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

.

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

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

→ cC3 (i.e. replace 3^rd column by ‘c’ times 3^rd 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 1^st column by addition of 1^st, 2^nd and 3^rd column)

Expanding determinant along 1^st column

∴ LHS = (1 + a² + b² + c²) = RHS

∴