Evaluate `|[1,x,y],[1,x+y,y],[1,x,x+y]|`

Evaluate |[x,y,x+y],[y,x+y,x],[x+y,x,y]|

The value of |[[1,x,y+z],[1,y,z+x],[1,z,x+y]]| is

The value of the det. |[x+y,y+z,z+x],[z,x,y],[1,1,1]| is

Use properties of determinants ot evaluate: {:|(x+y,y+z,z+x),(z,x,y),(1,1,1)|

Prove that |(1,x,x^2),(1,y,y^2),(1,z,z^2)| = (x-y)(y-z)(z-x)

Prove that: {:|(1,x,x^3),(1,y,y^3),(1,z,z^3)| = (x-y)(y-z)(z-x)(x+y+z)

If A(x_(1), y_(1)), B(x_(2), y_(2)) and C (x_(3), y_(3)) are the vertices of a Delta ABC and (x, y) be a point on the internal bisector of angle A, then prove that b|(x,y,1),(x_(1),y_(1),1),(x_(2),y_(2),1)|+c|(x,y,1),(x_(1),y_(1),1),(x_(3),y_(3),1)|=0 where, AC = b and AB = c.

If [[e^x,e^y],[e^y,e^x]] = [[1,1],[1,1]] then (x,y) = ___________

Find x and y, if 2[[1,3],[0,x]]+[[y,0],[1,2]] = [[5,6],[1,8]]