Prove that
`abs[[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]]=(a+b+c)^3`

Using properties of determinants prove the following. abs[[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]]=(a+b+c)^3

Using properties of determinants prove the following. abs[[b+c,a,a],[b,c+a,b],[c,c,a+b]]=4abc

Prove that abs[[(b+c)^2,a^2,a^2],[b^2,(c+a)^2,b^2],[c^2,c^2,(a+b)^2]] =2abc(a+b+c)^3

If Delta=[(a-b-c,2a,2a),(2b,b-c-a,2b),(2c,2c,c-a-b)] Perform R_1rarrR_1+R_2+R_3 on Delta

prove that abs[[a,b,c],[a+2x,b+2y,c+2z],[x,y,z]]=0

If Delta=[(a-b-c,2a,2a),(2b,b-c-a,2b),(2c,2c,c-a-b)] Perform R_1rarrR_1+R_2+R_3 on Delta . Take (a+b+c) common from R_1 & rewrite Delta .

using properties of determinants, prove that abs[[1,a,a^2],[1,b,b^2],[1,c,c^2]]=(a-b)(b-c)(c-a) .

Using properties of determinants, show that abs[[a,a^2,b+c],[b,b^2,c+a],[c,c^2,a+b]]=(b-c)(c-a)(a-b)(a+b+c)

Show that abs[[1,a,a^2],[1,b,b^2],[1,c,c^2]]=(a-b)(b-c)(c-a)

Show that Delta=abs[[-a^2,ab,ac],[ba,-b^2,bc],[ac,bc,-c^2]]=4a^2b^2c^2