(w)|[b+c,a,a],[b,c+a,b],[c,c,a+b]|

If a , b , c are real numbers, prove that |[a,b,c],[b,c,a],[c,a,b]|=-(a+b+c)(c+b w+c w^2)(a+b w^2+c w), where w is a complex cube root of unity.

Prove that: |[a+b, b+c, c+a],[b+c,c+a,a+b],[c+a,a+b,b+c]|=2|[a,b,c],[b,c,a],[c,a,b]|

Evaluate the following determinants: [[a-b,b-c,c-a],[b-c,c-a,a-b],[c-a,a-b,b-c]]

If alpha=|(a+b,b+c,c+a),(b+c,c+a,a+b),(c+a,a+b,b+c)|,beta=|(a,b,c),(b,c,a),(c,a,b)| , then

Show that |{:(b+c,c+a,a+b),(c+a,a+b,b+c),(a+b,b+c,c+a):}|=2|{:(a,b,c),(b,c,a),(c,a,b):}|

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

|(a,b,c),(a-b,b-c,c-a),(b+c,c+a,a+b)|=