Prove that
`|(-a,b,c),(a,-b,c),(a,b,-c)| = 4abc`

Prove that : |{:(a+b,b,c),(b+c,c,a),(c+a,a,b):}|=3 abc-a^(3)-b^(3)-c^(3)

Prove that : (i) |{:(a,c,a+c),(a+b,b,a),(b,b+c,c):}|=2 abc (ii) Prove that : |{:(a^(2),bc,ac+c^(2)),(a^(2)+ab,b^(2),ac),(ab,b^(2)+bc,c^(2)):}|=4a^(2)b^(2)c^(2)

Prove that |{:(b+c,c,b),(c,c+a,a),(b,a,a+b):}| =4 abc.

Prove that |(a+b, b, c), (b+c, c, a), (c+a, a, b)| = 3abc - a^3-b^3-c^3.

Prove that |(a+b,b+c,c+a),(b+c,c+a,a+b),(c+a,a+b,b+c)|=- 2 (a^3+ b^3+c^3-3abc)

Using properties of determinants, show that |(b+a,a,a),(b,c+a,b),(c,c,a+b)|=4abc

Prove that ((a+b+c)(b+c-a)(c+a-b)(a+b-c))/(4b^(2)c^(2))=sin^(2)A

Prove that: |[-2a, a+b,a+c],[ b+a,-2b,b+c],[c+a, c+b,-2c]|=4(a+b)(b+c)(c+a)

Prove that |[a,b,c] , [a^2,b^2,c^2] , [a^3,b^3,c^3]|= abc(a-b)(b-c)(c-a)

Using properties of determinants,prove the b+c,a,ab,c+a,bc,c,a+b]|=4abc