`|[a+b,a,b],[a,a+c,c],[b,c,b+c]|`is equal to

|{:(a,b,c),(b,c,a),(c,a,b):}| is equal to

Let a ,b ,c be positive and not all equal. Show that the value of the determinant |[a, b, c],[b, c, a],[ c, a ,b]| is negative.

Prove the identities: |[a, b-c,c-b],[ a-c, b, c-a],[ a-b,b-a, c]| =(a+b-c)(b+c-a)(c+a-b)

If A={a,b},B={c,d},C={d,e} , then {(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)} is equal to

If A={a,b},B={c,d},C={d,e} , then {(a,c),(a,d),(a,e),(b,c),(b,d),(b,e)} is equal to

If a, b and c are real numbers, and Delta=|[b+c,c+a,a+b],[c+a,a+b,b+c],[a+b,b+c,c+a]|=0 . Show that either a+b+c=0 or a=b=c

Prove: |[b+c ,a-b ,a], [c+a, b-c ,b], [a+b, c-a, c]|=3a b c-a^3-b^3-c^3

Prove the identities: |[a, b, c],[ a-b,b-c,c-a],[ b+c,c+a, a+b]|=a^3+b^3+c^3-3a b c

The product of all values of t , for which the system of equations (a-t)x+b y+c z=0,b x+(c-t)y+a z=0,c x+a y+(b-t)z=0 has non-trivial solution, is (a) |[a, -c, -b], [-c, b, -a], [-b, -a, c]| (b) |[a, b, c], [b, c, a], [c, a, b]| (c) |[a, c, b], [b, a, c], [c, b, a]| (d) |[a, a+b, b+c], [b, b+c, c+a], [c, c+a, a+b]|

Show that |[b-c,c-a, a-b],[ c-a, a-b,b-c],[ a-b,b-c,c-a]| = 0 .