# Matrices (transpose with unknowns)

• Apr 5th 2008, 02:38 AM
mathsstudent8
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

• Apr 5th 2008, 04:00 AM
mr fantastic
Quote:

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 $\displaystyle 5 k x_1^2 + 10 x_1 x_2 + x_2^2 > 0$ for all non-zero values of $\displaystyle x_1$ and $\displaystyle x_2$.

In other words, $\displaystyle k > -\left( \frac{10x_1 x_2 + x_2^2}{5 x_1^2} \right)$ for all non-zero values of $\displaystyle x_1$ and $\displaystyle x_2$.

It's not hard to see that the least upper bound of $\displaystyle -\left( \frac{10x_1 x_2 + x_2^2}{5 x_1^2} \right)$ is zero. Therefore .....

Alternatively, you could think of $\displaystyle 5 k x_1^2 + 10 x_1 x_2 + x_2^2 = 0$ as representing a conic section ......