Using properties of determinants prove that:
[C3’ = 2C3]
[C1’ = C1 - C3]
[C1’ = C1 + C2]
[C1’ = C1/(l2 + m2 + n2)]
[transforming row and column]
[C1’ = C1 - C2 & C2’ = C2 - C3]
[C1’ = C1/(l - m) & R2’ = C2/(l - m)]
= (l2 + m2 + n2)(l - m)(m - n){0 + 0 - l(l + m) + n(m + n)} [expansion by first row]
= (l2 + m2 + n2)(l - m)(m - n){0 + 0 - l(l + m) + n(m + n)}
= (l2 + m2 + n2)(l - m)(m - n)( - l2 - ml + mn + n2)
= (l2 + m2 + n2)(l - m)(m - n){(n2 - l2) + m(n - l)}
= (l2 + m2 + n2)(l - m)(m - n)(n - l)(l + m + n)