Without expanding at any stage, prove that `|(0,99,-998),(-99,0,997),(998,-997,0)|=0`

witout expanding at any stage Prove that |{:(0,,a,,-b),(-a,,0,,-c),(b,,c,,0):}| =0

Without expanding at any state, prove that |(1,bc,a(b+c)),(1,ca,b(c+a)),(1,ab,c(a+b))|=0

Without expanding at any state, prove that |((1)/(a),a,bc),((1)/(b),b,ca),((1)/(c ),c,ab)|=0

Without expanding at any stage, prove that |{:(1,omega,omega^(2)),(omega,omega^(2),1),(omega^(2),1,omega):}| = 0

Without expanding the determinant prove that: |(0, p-q, p-r), (q-p,0, q-r),(r-p, r-q, 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 |(2,7,65), (3,8,75), (5,9,86)|=0

Using the property of determinants and without expanding in questions 1 to 7 prove that , |{:(x,a,x+a),(y,b,y+b),(z,c,z+c):}|=0

If alpha,beta "and" gamma are real number without expanding at any stage prove that |{:(1,cos(beta-alpha),cos(gamma-alpha)),(cos(alpha-beta),1,cos(gamma-beta)),(cos(alpha-gamma),cos(beta-gamma),1):}| =0.

Without expanding, prove that Delta=|(x+y,y+z,z+x),(z,x,y),(1,1,1)|=0