" 11.Prove that "|[x+a,b,c],[a,x+b,c],[a,b,x+c]|=x^(2)(x+a+b+c)

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)

S.T abs[[x+a,b,c],[a,x+b,c],[a,b,x+c]]=x^2(x+a+b+c)

The determinant |[y^2,-x y, x^2],[a, b, c],[ a ', b ', c ']| is equal to a. |[b x+a y, c x+b y],[ b^(prime)x+a ' y, c^(prime)x+b ' y]| b. |[a x+b y, b x+c y],[ a^(prime)x+b ' y, b ' x+c ' y]| c. |[b x+c y, a x+b y],[ b^(prime)x+c ' y, a^(prime)x+b ' y]| d. |[a x+b y, b x+c y],[ a^(prime)x+b ' y, b^(prime)x+c ' y]|

Prove that |[a x-b y-c z, a y+b x, c x+a z], [a y+b x, b y-c z-a x, b z+c y],[c x+a z, b z+c y, c z-a x-b y]|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

Prove that |[a x-b y-c z, a y+b x, c x+a z], [a y+b x, b y-c z-a x, b z+c y],[c x+a z, b z+c y, c z-a x-b y]|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

if x^(3) =1,prove : |{:(a,b,c),(b,c,a),(c,a,b):}|=(a+bx+cx^2)|{:(1,b,c),(x^2,c,a),(x,a,b):}|

Prove that: |a b a x+b y b c b x+c y a x+b y b x+c y0|=(b^2-a c)(a x^2+2b x y+c y^2) .

Prove that (a+b+c+x) is a factor of |(x+a, b,c),(b,x+c,a),(c,a,x+b)|

Prove that (x^(b-c))^(a)(x^(a-b))^(c)(x^(c-a))^(b)=1

Prove that |a x-b y-c z a y+b x c x+a z a y+b x b y-c z-a x b z+c y c x+a z b z+c y c z-a x-b y|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot