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

Show that abs[[a+b,b+c,c+a],[b+c,c+a,a+b],[c+a,a+b,b+c]]=2abs[[a,b,c],[b,c,a],[c,a,b]]

Using properties of determinants, show that abs[[a,a^2,b+c],[b,b^2,c+a],[c,c^2,a+b]]=(b-c)(c-a)(a-b)(a+b+c)

prove that abs[[a,b,c],[a+2x,b+2y,c+2z],[x,y,z]]=0

Show that abs[[1,a,a^2],[1,b,b^2],[1,c,c^2]]=(a-b)(b-c)(c-a)

Compute the following: i) [[a, b],[ -b, a]]+[[a, b],[ b , a]] . ii) [[a^2+b^2 , b^2+c^2], [a^2+c^2 , a^2+b^2]]+[[2 a b , 2 b c],[ -2 a c, -2 a b]] iii) [[-1 , 4 ,-6] ,[ 8, 5 , 16],[ 2 , 8 , 5]]+[[12, 7 ,6 ],[ 8, 0, 5],[ 3, 2, 4]] iv) [[cos ^2 x ,. sin ^2 x],[ sin ^2 x , cos ^2 x]]+[[sin ^2 x, cos ^2 x ],[ cos ^2 x ,sin ^2 x]] .

|{:(a+x, b, c), (a, b+y, c), (a, b, c+z):}| is equal to :

Prove that Delta=|[a+b x, c+d x, p+q x],[ a x+b, c x+d, p x+q],[ u, v, w]|=(1-x^2)|[a , c, p],[ b, d, q],[ u, v, w]| .

The value of the determinant |(bc,ac,ab),(a,b,c),(a^(3),b^(3),c^(3))| is a)(a + b + c) b)a + b + c -ab c) (a^(2) - b^(2))(b^(2) - c^(2)) (c^(2) - a^(2)) d) a^2+b^2+c^2

Determinant Delta=|(a^(2)+x,ab,ac),(ab,b^(2)+x,bc),(ac,bc,c^(2)+x)| is divisible by a)abc b) x^(2) c) x^(3) d)None of these

If y=|[f(x), g(x), h(x)],[ l, m , n],[ a , b , c]| prove that (dy/dx)=|[f'(x), g'(x), h'(x)],[ l, m , n],[ a , b , c]|