17. The quadratic form Q(x, y, z) is given by Q(x,y,z) = 5x² + 3xy +3y2 + 2xz + 3yz + tz2 %3D Let A be the symmetric matrix so that Q(x, y, z) = (X y _z)A(y). There is one value of t for which Nul A is nontrivial, and for this value of t, there is a vector ( b] for which Nul A = span For this value of t, give the value of +t, unless this value b. a+b does not exist. In this case, record your answer as 999.
17. The quadratic form Q(x, y, z) is given by Q(x,y,z) = 5x² + 3xy +3y2 + 2xz + 3yz + tz2 %3D Let A be the symmetric matrix so that Q(x, y, z) = (X y _z)A(y). There is one value of t for which Nul A is nontrivial, and for this value of t, there is a vector ( b] for which Nul A = span For this value of t, give the value of +t, unless this value b. a+b does not exist. In this case, record your answer as 999.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 17: Quadratic Form and Null Space**
The quadratic form \( Q(x, y, z) \) is defined by:
\[ Q(x, y, z) = 5x^2 + 3xy + 3y^2 + 2xz + 3yz + tz^2 \]
Let \( A \) be the symmetric matrix such that:
\[ Q(x, y, z) = (x \; y \; z) A \begin{pmatrix} x \\ y \\ z \end{pmatrix} \]
There exists one value of \( t \) for which the null space of \( A \) (denoted as \(\text{Nul } A\)) is nontrivial. For this value of \( t \), there is a vector
\[
\begin{pmatrix} a \\ b \\ c \end{pmatrix}
\]
such that:
\[
\text{Nul } A = \text{span} \left\{ \begin{pmatrix} a \\ b \\ c \end{pmatrix} \right\}
\]
For this particular value of \( t \), determine the value of \( \frac{a+b}{c} + t \), unless this value does not exist. If it does not exist, record your answer as 999.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85682c27-480f-4c52-934f-0044190c2b1a%2Fe362c289-8ad2-4035-ba57-cf2e10881edb%2Frw89fhq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 17: Quadratic Form and Null Space**
The quadratic form \( Q(x, y, z) \) is defined by:
\[ Q(x, y, z) = 5x^2 + 3xy + 3y^2 + 2xz + 3yz + tz^2 \]
Let \( A \) be the symmetric matrix such that:
\[ Q(x, y, z) = (x \; y \; z) A \begin{pmatrix} x \\ y \\ z \end{pmatrix} \]
There exists one value of \( t \) for which the null space of \( A \) (denoted as \(\text{Nul } A\)) is nontrivial. For this value of \( t \), there is a vector
\[
\begin{pmatrix} a \\ b \\ c \end{pmatrix}
\]
such that:
\[
\text{Nul } A = \text{span} \left\{ \begin{pmatrix} a \\ b \\ c \end{pmatrix} \right\}
\]
For this particular value of \( t \), determine the value of \( \frac{a+b}{c} + t \), unless this value does not exist. If it does not exist, record your answer as 999.
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