Show that `|1+a1 1 1 1+b1 1 1 1+c|=a b c(1+1/a+1/b+1/c)=a b c+b c+c a+a b`

Prove that |[1+a,1, 1], [1,1+b,1], [1, 1, 1+c]|=a b c(1+1/a+1/b+1/c)=a b c+b c+c a+a b

Prove that |1+a1 1 1 1 1+b1 1 1 1 1+c1 1 1 1 1+d|=a b c d(a+1/a+1/b+1/c+1/d)dot Hence find the value of the determinant if a ,b ,c , d are the roots of the equation p x^4+q x^3+r x^2+s x+t=0.

Show that |1a b_c1b c+a1c a+b|=0

Using properties of determinant show that: |[1 , a , bc] , [1 , b , ca] , [1 , c , a b]|=(a-b)(b-c)(c-a)

If a, b, c are positive real numbers then show that (i) (a + b + c) ((1)/(a) + (1)/(b) +(1)/(c)) ge 9 (ii) (b+c)/(a) +(c +a)/(b) + (a+b)/(c) ge 6

Show that |[1,b c, a(b+c)],[1,c a, b(c+a)],[1,a b ,c(a+b)]|=0 .

If the value of the determinant |(a,1, 1) (1,b,1) (1,1,c)| is positive then a. a b c >1 b. a b c> -8 c. a b c -2

Using properties of determinants, show that |1 a a^2 -b c 1 b b^2 -c a 1 c c^2 -a b|=0

If the origin lies in the acute angle between the lines a_1 x + b_1 y + c_1 = 0 and a_2 x + b_2 y + c_2 = 0 , then show that (a_1 a_2 + b_1 b_2) c_1 c_2 lt0 .

By using properties of determinants. Show that: |[a^2+1,a b, a c],[ a b,b^2+1,b c],[c a, c b, c^2+1]|=(1+a^2+b^2+c^2)