Prove: `|(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)|`

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):}| .

5. Using the properties of determinants, 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]|

Prove: |(1,a, b c),(1,b ,c a),(1,c ,a b)|=|(1,a ,a^2),( 1,b,b^2),( 1,c,c^2)|

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)

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

The value of |(a-b,b+c,a),(b-a,c+a,b),(c-a,a+b,c)| is

Show without expanding at any stage 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]|

The value of |{:(a- b, b- c, c - a),(b-c,c-a,a-b),(c-a,a-b,b-c):}| =