prove that `|{:(a^(2)+1,,ab,,ac),(ab,,b^(2)+1,,bc),(ac,,bc,,c^(2)+1):}| = 1+ a^(2) +b^(2)+c^(2).`

Using properties of determinants, prove that : |{:(a^(2)+1,ab,ac),(ba,b^(2)+1,bc),(ca,cb,c^(2)+1):}|=a^(2)+b^(2)+c^(2)+1

What is |{:(-a^(2),ab,ac),(ab,-b^(2),bc),(ac,bc,-c^(2)):}| equal to ?

Using properties of determinant prove that |(a^(2)+1, ab, ac),(ab, b^(2)+1, bc),(ca, cb,c^(2)+1)|=(1+a^(2)+b^(2)+c^(2)) .

Prove that : (i) |{:(a,c,a+c),(a+b,b,a),(b,b+c,c):}|=2 abc (ii) Prove that : |{:(a^(2),bc,ac+c^(2)),(a^(2)+ab,b^(2),ac),(ab,b^(2)+bc,c^(2)):}|=4a^(2)b^(2)c^(2)

Prove that |{:(a,,a^(2),,bc),(b ,,b^(2),,ac),( c,,c^(2),,ab):}| = |{:(1,,1,,1),(a^(2) ,,b^(2),,c^(2)),( a^(3),, b^(3),,c^(3)):}|

If a, b, c are non zero complex numbers satisfying a^(2) + b^(2) + c^(2) = 0 and |(b^(2) + c^(2),ab,ac),(ab,c^(2) + a^(2),bc),(ac,bc,a^(2) + b^(2))| = k a^(2) b^(2) c^(2) , then k is equal to

Using properties of determinants, prove the following |(a^2+1,ab,ac),(ab,b^2+1,bc),(ca,cb,c^2+1)|=1+a^2+b^2+c^2 .

Using properties of determinants, prove the following |(a^2,ab,ac),(ab,b^2+1,bc),(ca,cb,c^2+1)|=1+a^2+b^2+c^2 .

Prove the following by multiplication of determinants and power cofactor formula |{:(0,c,b),(c,0,a),(b,a,0):}|^(2)=|{:(b^(2)+v^(2),ab,ac),(ab,c^(2)+a^(2),bc),(ac,bc,a^(2)+b^(2)):}| =|{:(-a^(2),ab,ac),(ab,-b^(2),bc),(ac,bc,-c^(2)):}|=4a^(2)b^(2)c^(2)

Prove the identities: |{:(b^(2)+c^(2),,ab,,ac),(ab,,c^(2)+a^(2),,bc),(ca,,bc,,a^(2)+b^(2)):}|=4a^2b^2c^2