Let A be the following symmetric matrix

A =

[4 2 -2 2

2 2 1 1

-2 1 14 -1

2 1 -1 x]

find the values of x , such that A is positive definite.

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- Oct 26th 2008, 07:59 AMtinngpositive definite matrix
Let A be the following symmetric matrix

A =

[4 2 -2 2

2 2 1 1

-2 1 14 -1

2 1 -1 x]

find the values of x , such that A is positive definite. - Oct 26th 2008, 09:16 AMmathemanyakanswer
a

**symmetric matrix**is a square matrix,*A*, that is equal to its transpose http://upload.wikimedia.org/math/a/f...b50df7fdfd.png and An*n*×*n*real symmetric matrix*M*is*positive definite*if*z*T*Mz*> 0 for all non-zero vectors*z*with real entries (i.e.*z*∈**R***n*), where*z*T denotes the transpose of*z*. - Oct 26th 2008, 09:57 AMLaurent
There is a theorem telling that $\displaystyle A=(a_{ij})_{1\leq i,j\leq n}$ (here, $\displaystyle n=4$) is positive definite if, and only if, for $\displaystyle k=1,\ldots,n$ the determinant of $\displaystyle A_k=(a_{ij})_{1\leq i,j\leq k}$ is positive.

If you know this, then it is easy. The determinants of $\displaystyle A_1, A_2, A_3$ are positive, they do not depend on $\displaystyle x$, and the determinant of $\displaystyle A_4=A$ is found to be $\displaystyle 36(x-1)$, hence it is positive iff $\displaystyle x>1$. So $\displaystyle A$ is positive definite iff $\displaystyle x>1$.