# Thread: Matrices (transpose with unknowns)

1. ## Matrices (transpose with unknowns)

Matrix A=
5k 1
9 1

calculate (transpose(x)*A*x ) where x=
x1
x2
find the set of values for k where (transpose(x)*A*x ) > 0 for all non zero x in R^2

I am having real difficulty on this one, as when i did this calculation i got:

5kx1^2 + 10x1x2 + x2^2

2. Originally Posted by mathsstudent8
Matrix A=
5k 1
9 1

calculate (transpose(x)*A*x ) where x=
x1
x2
find the set of values for k where (transpose(x)*A*x ) > 0 for all non zero x in R^2

I am having real difficulty on this one, as when i did this calculation i got:

5kx1^2 + 10x1x2 + x2^2

I don't see what the confusion is ...... You require the values of k such that $5 k x_1^2 + 10 x_1 x_2 + x_2^2 > 0$ for all non-zero values of $x_1$ and $x_2$.
In other words, $k > -\left( \frac{10x_1 x_2 + x_2^2}{5 x_1^2} \right)$ for all non-zero values of $x_1$ and $x_2$.
It's not hard to see that the least upper bound of $-\left( \frac{10x_1 x_2 + x_2^2}{5 x_1^2} \right)$ is zero. Therefore .....
Alternatively, you could think of $5 k x_1^2 + 10 x_1 x_2 + x_2^2 = 0$ as representing a conic section ......