MUST SHOW SOLUTION 4 a 5 pts Find a 3 x 3 matrix A so that t

MUST SHOW SOLUTION 4. (a) (5 pts) Find a 3 x 3 matrix A so that the quadratic polynomial h of x, y, z defined by h(x, y, z) = x^2 + y^2 + z^2 - 2xy -4yz - 6xz can be written as h(u) = u^TAu where u = [ ] (b) (5 pts) Find all possible values of k set membership R so that [ ] [ ] [ ]= 0.

Solution

h(x,y,z)=x^2+y^2+z^2-2xy-4yz-6xz

so quadratic polynomial can be written as matrix form as

H=[1 -1 -3

-1 1 -2

-3 -2 1]

2) [1 2 k] [1 -2 0 [1

-2 0 3 2

0 3 1] k] =0

=[-3 -2+3k 6+k] [1

2

k] =0

so k^2++12k-7=0

so k=-12+root(116)/2 or  k=-12-root(116)/2

so k=-6+root(29) or k=6-root(29)