Without expanding, prove that `|a+b x c+dx 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|` .

Without expanding, prove that |(a+b x, c+dx, 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)| .

Without expanding, prove that |a b c x y z p q r|=|x y z p q r a b c|=|y b q x a p z c r|

Without expanding, prove that |a b c x y z p q r|=|x y z p q r a b c|=|y b q x a p z c r|

Without expanding, prove that |a b c x y z p q r|=|x y z p q r a b c|=|y b q x a p z c r|

Without expanding prove that , |{:(a+bx,c+dx,p+qx),(ax+b,cx+d,px+q),(u,v,w):}|=(1-x^2)|{:(a,c,p),(b,d,q),(u,v,w):}|

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]| .

Prove that |(a+bx ,c+dx,p+qx),(ax + b, cx +d, px +q),(u,v,w)|= (1- x^3) |(a,c,p),(b,d,q),(u,v,w)|

Using the property of determinants and without expanding, prove that: |b+c q+r y+z c+a r+p z+x a+b p+q x+y|=2|a p x b q y c r z|

Using the property of determinants and without expanding, prove that: |b+c q+r y+z c+a r+p z+x a+b p+q x+y|=2|a p x b q y c r z|

Prove that /_\ |[a+bx, c+dx, p+qx],[-ax+b, cx+d, px+q],[u,v,w]|=(1-x^2) [[a,c,p],[b,d,q],[u,v,w]]