Prove that: `|[0,a,-b],[-a,0,-c],[ b, c,0]|=0` .

Find the number of real root of the equation |[0,x-a, x-b],[ x+a,0,x-c],[ x+b, x+c,0]|=0,a!=b!=c and b(a+c)> a c

If a , b , c , are roots of the equation x^3+p x+q=0, prove that |[a ,b, c],[ b ,c, a],[ c, a, b]|=0

Using the property of determinants and without expanding, prove that: |0a-b-a0-c b c0|=0

Prove that : |{:(0,a-b,a-c),(b-a,0,b-c),(c-a,c-b,0):}|=0

Using the property of determinants and without expanding, prove that: |[a-b,b-c,c-a],[ b-c,c-a ,a-b ],[c-a ,a-b,b-c]|=0

Without expanding or evaluating show that |[0 , b-a , c-a] , [a-b , 0 , c-b] , [a-c , b-c , 0]|=0

Without expanding or evaluating show that |(0,b-a, c-a),( a-b,0,c-b),( a-c ,b-c,0)|=0 .

Using the property of determinants and without expanding, prove that: |[1,b c, a(b+c)],[1,c a, b(c+a)],[1,a b, c(a+b)]|=0

If a ,b ,c are different, then the value of |[0,x^2-a, x^3-b],[ x^2+a,0,x^2+c],[ x^4+b, x-c,0]| is a. c b. a c. b d. 0

Solve the equation |[a-x, c, b], [ c, b-x, a], [b, a, c-x]|=0 where a+b+c!=0.