Prove that: `|[a+b+2c,a,b],[c,b+c+2a,b],[c,a,c+a+2b]|= 2(a+b+c)^3`

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

Show that: |[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]|=(a+b+c)^3

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: |[a-b-c,2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]|=(a+b+c)^3

Prove that: |[a^2+1,ab,ac],[ab,b^2+1,bc],[ac,bc,c^2+1]|=|[a^2+1,b^2,c^2],[a^2,b^2+1,c^2],[a^2,b^2,c^2+1]|=1+a^2+b^2+c^2

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

Without expanding the determinant, prove that |[a,a^2,bc],[b,b^2,ca],[c,c^2,ab]| = |[1,a^2,a^3],[1,b^2,b^3],[1,c^2,c^3]|

Using the properties of determinant, show that : |[1,a+b,a^2+b^2],[1,b+c,b^2+c^2],[1,c+a,c^2+a^2]| = (a-b)(b-c)(c-a)

Using the properties of determinants, prove that : |[[x+a,b,c],[a,x+b,c],[a,b,x+c]]| = x^2(x+a+b+c)