The signature of the quadractic Form Q is$ (+,+,0)$? Which of the following statment is true

Question

let $$A=\begin{pmatrix}1&2 &0\\0&0&-2\\0&0&1\end{pmatrix}.$$ and define for $x,y, z$ $\in R$

$Q(x,y,z) $= $(x,y,z) A \begin{pmatrix}x\\y\\z\end{pmatrix}$

which of the following satement is True ?

a) The matrix of second order partial derivatives of the quadratics form Q is $2A$

b) The ranks of the quadratics form $Q$ is $2$

c) The signature of the quadractic Form Q is$ (+,+,0)$

d) The Quadratics Form $Q$ takes the Value 0 for some non zero vectors $(x,y,z)$

My attempts : Matrix A has ranks 2 and it has 2 postive sigen, and one is 0,,,,in my thinking option 2 and and option 3 is correct...

is my answer is correct..??

Any hints/solution will be appreciated,,,,

i would be th thanksfuls..

score 2 · Accepted Answer · answered Jun 18 '18 at 05:55

2

We have that $A$ can be divided as the sum of a symmetric and a skew part

$$A=B+C=\begin{pmatrix}1&2 &0\\0&0&-2\\0&0&1\end{pmatrix}=\begin{pmatrix}1&1 &0\\1&0&-1\\0&-1&1\end{pmatrix}+\begin{pmatrix}0&1 &0\\-1&0&-1\\0&1&0\end{pmatrix}$$

with $x^TCx=0$ and for the symmetric part $B$ we have

$\det(1)=1>0$
$\det\left(\begin{smallmatrix}1&1 \\1&0\end{smallmatrix}\right)=-1<0$
$\det(B)=-2<0$

therefore the signature for $B$ and thus for $A$ is $(+,+,-)$.

answered Jun 18 '18 at 05:55

user

154,566

what about option 4 ?? pliz tell me gimusi – jasmine Jun 18 '18 at 06:03
1

We can exclude now $b$ and $c$, $a$ also can be excluded because $A$ is not symmetric. Threfore oprion $d$ is the correct one. Can you find a vector $v\neq 0$ such that $Av=0$? – user Jun 18 '18 at 06:05
thanks a lots @Gimusi – jasmine Jun 18 '18 at 06:07
1

You are welcome! What about the vector $v$? – user Jun 18 '18 at 06:11
1

To complete the problem you should find such vector $v$. Let try to find it, it is not a difficult task. – user Jun 18 '18 at 06:44
v= (0,0,1) is it correct ? – jasmine Jun 18 '18 at 06:55
1

Let try,$ A(0,0,1)=(0,-2,1)\neq 0$. Recall that it suffices to solve $Av=0$. – user Jun 18 '18 at 06:57

The signature of the quadractic Form Q is$ (+,+,0)$? Which of the following statment is true

1 Answers1

Linked