Show that `|{:(a^(2)+x^(2),ab-cx,ac+bx),(ab+cx,b^(2)+x^(2),bc-ax),(ac-bx,bc+ax,c^(2)+x^(2)):}|=|{:(x,c,-b),(-c,x,a),(b,-a,x):}|^(2)`.

Statement I If A=[(a^2+x^2,ab-cx,ac+bx),(ab+cx,b^2+x^2,bc-ax),(ac-bx,bc+ax,c^2+x^2)] and B[(x,c,-b),(-c,x,a),(b,-a,x)] , then |A|=|B|^2 Statement II A^c is cofactor of a square matrix A of order n, then |A^c|=|A|^(n-1)

Statement I If A=[(a^2+x^2,ab-cx,ac+bx),(ab+cx,b^2+x^2,bc-ax),(ac-bx,bc+ax,c^2+x^2)] and B[(x,c,-b),(-c,x,a),(b,-a,x)] , then |A|=|B|^2 Statement II A^c is cofactor of a square matrix A of order n, then |A^c|=|A|^(n-1)

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

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

The determinant Delta=|{:(,a^(2)(1+x),ab,ac),(,ab,b^(2)(1+x),(bc)),(,ac,bc,c^(2)(1+x)):}| is divisible by

"If "f(x)=|{:(x+a^(2),ab,ac),(ab, x+b^(2),bc),(ac,bc, x+c^(2)):}|," then prove that " f'(x)=3x^(2)+2x(a^(2)+b^(2)+c^(2)) .

"If "f(x)=|{:(x+a^(2),ab,ac),(ab, x+b^(2),bc),(ac,bc, x+c^(2)):}|," then prove that " f'(x)=3x^(2)+2x(a^(2)+b^(2)+c^(2)) .

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

|[x^2+a^2,ab,ac] , [ab,x^2+b^2,bc] , [ac,bc,x^2+c^2]|=