Show that
`Delta=|[(y+z)^2, x y , z x],[ x y, (x+z)^2, y z],[ x z, y z, (x+y)^2]|=2 x y z(x+y+z)^3`

Given Delta=|(y+z,x,y),(z+x,z,x),(x+y,y,z)| Prove that Delta=(x+y+z)(x-z)^2 .

Given Delta=|(y+z,x,y),(z+x,z,x),(x+y,y,z)| Take (x+y+z) common from R_1 .

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

Given Delta=|(y+z,x,y),(z+x,z,x),(x+y,y,z)| Apply R_1rarrR_1+R_2+R_3 .

Find the values of x, y,z from the following equations i) [[4, 3 ], [x, 5]]=[[y , z], [1 , 5]] ii) [[x+y , 2] , [5+z, x y]]=[[6 , 2] , [5, 8]] iii) [[x+y+z], [x+z] , [y+z]]=[[9] , [5 ] ,[7]]

If the equation of the sphere through the circle and the plane x ^(2) + y ^(2) + z^(2) = 5, 2x + 3y + 4z = 5 and through the origin is x ^(2) + y ^(2) + z ^(2) - 2x - 3y - 4z + c =0 .Then value of c is

If a,b,c,d,e and f are in GP, then the value of determinant |{:(a ^(2) , d ^(2) , x), ( b ^(2) , e^(2), y) , (c^(2) , f^(2), z}| depends on a) x and y b) x and z c) y and z d) independent of x,y,z

Prove that |(x^(2),x^(2)-(y-z)^(2),yz),(y^(2),y^(2)-(z-x)^(2),zx),(z^(2),z^(2)-(x-y)^(2),xy)|=(x-y)(y-z)(z-x)(x+y+z)(x^(2) + y^(2) + z^(2)) .

By using properties of determinants, prove that |[x,x^2,yz],[y,y^2,zx],[z,z^2,xy]|=(x-y)(y-z)(z-x)(xy+yz+zx)

Let R be the relation on the set Z of all integers defined by R={(x, y) : x,y in Z,x-y is divisible by n}.Prove that i) (x, x) in R for all x in , Z ii) (x, y) in R implies that (y, x) in R for all x, y in Z iii) (x, y) in R and (y, z) in R implies that (x, z) in R for all x, y, z in Z