Show that the following determinant is a perfact square :
`|{:(1,a,a^2),(a^2,1,a),(a,a^2,1):}|`

Show that the following determinant is a perfect square |[1,a,a^2],[a^2,1,a],[a,a^2,1]|

Show that the determinant |{:(1,a,a^(2)),(a^(2),1,a),(a,a^(2),1):}| is a perfect square.

Express each of the following expressions as the sum of two squares: (1+x^(2))(1+y^(2))(1+z^(2))

By using properties of determinants , show that : {:|( 1,x,x^(2) ),( x^(2) ,1,x) ,( x,x^(2), 1) |:} =( 1-x^(3)) ^(2)

Express each of the following expressions as the sum of two squares: (1+x^(2))(1+y^(2))

Using determinant : show that the area of the triangle with vertices at (a^2,a^3),(b^2,b^3)and (c^2,c^3) is 1/2(a-b)(b-c)(c-a)(ab+bc+ca) square unit.

Evaluate the determinants in Exercises 1 and 2. {:|( 2,4),( -5,-1) |:}

Using determinant : show that the points (a+1,a) ,(a,a+1) and [(a+1)^2,-a^2] are collinear.

Without expanding show that the determinant |(1,a,a^2),(-1,2,4),(1,x,x^2)| has a factor (X-a)

The x-coordinates of the vertices of a square of unit area are the roots of the equation x^2-3|x|+2=0 . The y-coordinates of the vertices are the roots of the equation y^2-3y+2=0. Then the possible vertices of the square is/are (a)(1,1),(2,1),(2,2),(1,2) (b)(-1,1),(-2,1),(-2,2),(-1,2) (c)(2,1),(1,-1),(1,2),(2,2) (d)(-2,1),(-1,-1),(-1,2),(-2,2)