Prove that: `(a-b)^2=a^2-2ab+b^2`

Prove that: {:|(1+a^2-b^2,2ab,-2b),(2ab,1-a^2+b^2,2a),(2b,-2a,1-a^2-b^2)|

If a,b,c,d are in proportion, then prove that (a^2+ab+b^2)/(a^2-ab+b^2)=(c^2+cd+d^2)/(c^2-cd+d^2)

Prove that |axxb|^2 =a^2b^2 - (a.b)^2

Prove that |axxb|^2 =a^2b^2 - (a.b)^2

Prove that |axxb^2 =a^2b^2 - (a.b)^2

Using properties of determinant : Prove that |(a^(2), 2ab, b^(2)),(b^(2),a^(2),2ab),(2ab,b^(2),a^(2))| = (a^(3) + b^(3))^(2)

If a + b + c = 0 , then prove that (a+b)^2/(ab) + (b+c)^2/(bc) + (c+a)^2/(ca)=3

Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]| is divisible by a+b+c and find the quotient.

Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]| is divisible by a+b+c and find the quotient.

Prove that abs((a,b,c),(a^2,b^2,c^2),(bc,ca,ab))=(a-b)(b-c)(c-a)(ab+bc+ca)