Question

Using properties of determinants prove that 1 a cube is square 1 a square 1 is equals to 1 minus x cube whole square

Answer 1

Answer:

Step-by-step explanation:

i) L.H.S.=∣∣∣∣∣111abca2b2c2∣∣∣∣∣

Applying R1→R1−R2 and R2→R2−R3

=∣∣∣∣∣001a−bb−cca2−b2b2−c2c2∣∣∣∣∣

=∣∣∣∣∣001a−bb−cc(a−b)(a+b)(b−c)(b+c)c2∣∣∣∣∣

=(a−b)(b−c)∣∣∣∣∣00111c(a+b)(b+c)c2∣∣∣∣∣

=(a−b)(b−c)[b+c−a−b]

=(a−b)(b−c)(c−a)=R.H.S.

(ii) L.H.S.=∣∣∣∣1aa31bb31cc3∣∣∣∣

Applying C1→C1−C2andC2→C2−C3

=∣∣∣∣0a−ba3−b30b−cb3−c31cc3∣∣∣∣

=∣∣∣∣∣0a−b(a−b)(a2+ab+b2)0b−c(b−c)(b2+bc+c2)1cc3∣∣∣∣∣

=(a−b)(b−c)∣∣∣∣01a2+ab+b201b2+bc+c21cc3∣∣∣∣

=(a−b)(b−c)[b2+bc+c2−a2−ab−b2]

=(a−b)(b−c)[bc−ab+c2−a2]

=(a−b)(b−c)[b(c−a)+(c+a)(c−a)]

=(a−b)(b−c)(c−a)(a+b+c)=R.H.S.