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Let x = (r1.12) E R? and y = (y.2) E R². i) Find a symmetric matrix A e M2.2(R) such that x" Ax = 1 + 4ry + y*. %3D i) Find a diagonal matrix De M22(R) and a rotation matrix QE M22(R) such that x Ax = y" Dy, where y = Q"x. 5) Explain why the equation x' Ax = 1 is an equation of a hyperbola.

Let x = (r1.12) E R? and y = (y.2) E R². i) Find a symmetric matrix A e M2.2(R) such that x" Ax = 1 + 4ry + y*. %3D i) Find a diagonal matrix De M22(R) and a rotation matrix QE M22(R) such that x Ax = y" Dy, where y = Q"x. 5) Explain why the equation x' Ax = 1 is an equation of a hyperbola.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 32E
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3. а) Let x = (1.12) e R² and y = (1. 2) E R². i) Find a symmetric matrix A E M22(R) such that x" Ax = r + 4ry + y*. ii) Find a diagonal matrix De M22(R) and a rotation matrix Q € M2.2(R) such that x' Ax = y" Dy, where y = Q"x. iii) Explain why the equation x" Ax = 1 is an equation of a hyperbola. iv) Find the canonical (yı, 92) coordinates of the tangent vectors to the asymptotes of the hyperbola above. That is, find vectors uj and uz as shown on the fig 1. v) Find the original (1.12) coordinates of the tangent vectors to the asymptotes of the hyperbola above. That is, find vectors uj and uz as shown on the fig 1.
Transcribed Image Text:3. а) Let x = (1.12) e R² and y = (1. 2) E R². i) Find a symmetric matrix A E M22(R) such that x" Ax = r + 4ry + y*. ii) Find a diagonal matrix De M22(R) and a rotation matrix Q € M2.2(R) such that x' Ax = y" Dy, where y = Q"x. iii) Explain why the equation x" Ax = 1 is an equation of a hyperbola. iv) Find the canonical (yı, 92) coordinates of the tangent vectors to the asymptotes of the hyperbola above. That is, find vectors uj and uz as shown on the fig 1. v) Find the original (1.12) coordinates of the tangent vectors to the asymptotes of the hyperbola above. That is, find vectors uj and uz as shown on the fig 1.
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