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8 (a) Find an orthogonal matrix $P$ and a diagonal matrix $D$ such that $D = P^{-1}AP$, where $\begin{aligned} A = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 2 & 2 \\ 2 & 2 & 5 \end{bmatrix} \end{aligned}$ (b) Use Descartes' rule of signs to determine whether the matrix $A$ is positive definite, positive semidefinite, or neither. (c) Perform a change of variables so that the quadratic form $f(x, y) = 3x^2 - 2xy + 3y^2$ becomes diagonal. (d) Sketch the curve given by the equation $3x^2 + 4xy + 6y^2 = 1$. (e) Compute the complex dot product in $C^2$ for the vectors $\begin{bmatrix} 1 \\ i \end{bmatrix}$ and $\begin{bmatrix} 1 + i \\ 0 \end{bmatrix}$.

          8 (a) Find an orthogonal matrix $P$ and a diagonal matrix $D$ such that $D = P^{-1}AP$, where
$\begin{aligned} A = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 2 & 2 \\ 2 & 2 & 5 \end{bmatrix} \end{aligned}$
(b) Use Descartes' rule of signs to determine whether the matrix $A$ is positive definite, positive
semidefinite, or neither.
(c) Perform a change of variables so that the quadratic form $f(x, y) = 3x^2 - 2xy + 3y^2$ becomes
diagonal.
(d) Sketch the curve given by the equation
$3x^2 + 4xy + 6y^2 = 1$.
(e) Compute the complex dot product in $C^2$ for the vectors
$\begin{bmatrix} 1 \\ i \end{bmatrix}$ and $\begin{bmatrix} 1 + i \\ 0 \end{bmatrix}$.

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8 (a) Find an orthogonal matrix P and a diagonal matrix D such that D = P^-1AP, where
A =
< b m a t r i x >
(b) Use Descartes' rule of signs to determine whether the matrix A is positive definite, positive
semidefinite, or neither.
(c) Perform a change of variables so that the quadratic form f(x, y) = 3x^2 - 2xy + 3y^2 becomes
diagonal.
(d) Sketch the curve given by the equation
3x^2 + 4xy + 6y^2 = 1.
(e) Compute the complex dot product in C^2 for the vectors
< b m a t r i x > and < b m a t r i x >.

Added by Heather T.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Victor Salazar

02/23/2022

Step 1

The characteristic polynomial of A is $$det(A - \lambda I) = \begin{vmatrix} 2 - \lambda & 1 & 2 \\ 1 & 2 - \lambda & 2 \\ 2 & 2 & 5 - \lambda \end{vmatrix}$$ Expanding the determinant, we get $$(2 - \lambda) \begin{vmatrix} 2 - \lambda & 2 \\ 2 & 5 - \lambda Show more…

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(2) Find an orthogonal matrix P and a diagonal matrix D such D = P<sup>-1</sup>AP, where $$A = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 2 & 2 \\ 2 & 2 & 5 \end{bmatrix}$$ (b) Use Descartes' rule of signs to determine whether the matrix A is positive definite, positive semidefinite, or neither. (c) Perform a change of variables so that the quadratic form f(x,y) = 3x<sup>2</sup> - 2xy + 3y<sup>2</sup> becomes diagonal. (d) Sketch the curve given by the equation 3x<sup>2</sup> + 4xy + 6y<sup>2</sup> = 1. (e) Compute the complex dot product in C<sup>2</sup> for the vectors $$ \begin{bmatrix} 1 \\ i \end{bmatrix} $$ and $$ \begin{bmatrix} 1 + i \\ 0 \end{bmatrix} $$.

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