Evaluate the determinants `|{:(ax,y,z),(x,ay,z),(x,y,az):}|`

|{:(x,z,0),(0,y,y),(z,0,x):}|

if x+y+z = 0 then prove that |{:(x,y,z),(x^2,y^2,z^2),(y+z,z+x,x+y):}|=0

|{:(" "0," "x,y),(-x," "0,z),(-y,-z,0):}|=0

Find the values of x, y, z if the matrix A=[(0, 2y,z),(x,y,-z),(x,-y,z)] satisfy the equation A'A=I .

Prove that |{:(ax,,by,,cz),(x^(2),,y^(2),,z^(2)),(1,,1,,1):}|=|{:(a,,b,,c),(x,,y,,z),(yz,,xz,,xy):}|

If u=ax+by+cz , v=ay+bz+cx , w=az+bx+cy , then the value of |{:(a,b,c),(b,c,a),(c,a,b):}|xx|{:(x,y,z),(y,z,x),(z,x,y):}| is

By using properties of determinants , show that : {:|( x,x^(2) , yz) ,( y,y^(2) , zx ) ,( z , z^(2) , xy ) |:} =( x-y)(y-z) (z-x) (xy+yz+ zx)

Find the values of x ,\ y,\ z if the matrix A=[(0,2y,z), (x, y,-z ),(x,-y, z)] satisfy the equation A^T\ A=I_3 .

The determinant |[x p+y, x, y], [y p+z, y, z],[0,x p+y, y p+z]|=0 if (a) x y z are in AP (b) x y z are in GP (c) x y z are in HP (d) xy yz zx are in AP

|{:(a+x," "y," "z),(" "x,a+y," "z),(" "x," "y,a+z):}|=a^2(a+x+y+z)