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If a, b and c are all positive real, the...

If a, b and c are all positive real, then prove that minimum value of determinant
`|{:(a^2+1,ab,ac),(ab,b^2+1,bc),(ac,bc,c^2+1):}|` = `1+a^2+b^2+c^2`

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

Find the minimum value of abs[[a^2,bc,ac+c^2],[a^2+ab,b^2,ac],[ab,b^2+bc,c^2]]

Prove the following: [[-a^2,ab,ac],[ab,-b^2,bc],[ac,bc,-c^2]]=4a^2b^2c^2

Show that without expanding at any stage |{:(1/a,a^2,bc),(1/b,b^2,ca),(1/c,c^2,ab):}|=0

Using properties of determinants , prove that |{:(1,a,bc),(1,b,ca),(1,c,ab):}|=(a-b)(b-c)(c-a)

Without expanding the determinants 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 properties of determinants, prove the following abs{:(a^2, bc, ac +c^2 ),(a^(2) + ab, b^(2),ac ),(ab, b^(2) + bc,c^(2) ):}=4a^(2) b^(2) c^(2) .

show that |[a^2+x^2,ab ,ac],[ab,b^2+x^2,bc],[ac,bc,c^2+x^2]| is divisible x^4

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