Prove: `|a^2+1a b a c a bb^2+1b cc a c b c^2+1|=1+a^2+b^2+c^2`

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

By using properties of determinants. Show that: |[a^2+1,a b, a c],[ a b,b^2+1,b c],[c a, c b, c^2+1]|=(1+a^2+b^2+c^2)

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: |1a a^2-b c1bb^2-c a1cc^2-a b|=0

Prove that: |1a a^2-b c1bb^2-c a1cc^2-a b|=0

Prove: |[1,a^2+bc, a^3],[ 1,b^2+c a, b^3],[ 1,c^2+a b, c^3]|=-(a-b)(b-c)(c-a)(a^2+b^2+c^2)

Prove that |b c-a^2c a-b^2a b-c^2-b c+c a+a bb c-c a+a bb c+c a-a b(a+b)(a+c)(b+c)(b+a)(c+a)(c+b)|=3.(b-c)(c-a)(a-b)(a+b+c)(a b+b c+c a)

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 =|1 1 1 a b c b c+a^2a c+b^2a b+c^2|=2(a-b)(b-c)(c-a)

Prove that [[1+a^2+a^4, 1+a b+a^2b^2 ,1+a c+a^2c^2],[ 1+a b+a^2b^2, 1+b^2+b^4, 1+b c+b^2c^2],[ 1+a c+a^2c^2, 1+b c+b^2c^2, 1+c^2+c^2]]=(a-b)^2(b-c)^2(c-a)^2