( 7 ) to ( 2₁ ) ² -Y for M-1 makes sense geometrically. ? Find the inverse matrix of M and explain why your answer b) Let & be the line through the origin (0,0) that is parallel to the vector (Y³). Use matrix multiplication and your matrix from (a) to find the 2 × 2 matrix N x = ( ₁ ) y that reflects v = across the line l.
( 7 ) to ( 2₁ ) ² -Y for M-1 makes sense geometrically. ? Find the inverse matrix of M and explain why your answer b) Let & be the line through the origin (0,0) that is parallel to the vector (Y³). Use matrix multiplication and your matrix from (a) to find the 2 × 2 matrix N x = ( ₁ ) y that reflects v = across the line l.
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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Transcribed Image Text:(a) What is the 2 × 2 matrix M that reflects v = (3) across the r axis, i.e. takes
( #) to ( ² ) ?
-Y
for M-1 makes sense geometrically.
? Find the inverse matrix of M and explain why your answer
(b) Let & be the line through the origin (0,0) that is parallel to the vector
·(1³).
Use matrix multiplication and your matrix from (a) to find the 2 × 2 matrix N
that reflects v = ( ) across the line l.
Hints: (i) The matrix that rotates vectors counterclockwise by angle is
cos
sin
mº).
sin 0
cos
(ii) Reflecting across the line is equivalent to rotating the plane clockwise by an
angle that moves to the x axis, then applying a reflection across the r axis, then
rotating counterclockwise to move back where it started. You can use matrix
multiplication to build one matrix that performs these three steps in order.
(c) Find N-¹ in two ways: first, by applying the usual formula for the inverse of a
2 x 2 matrix to N, and second, by using the rule (ABC)-¹ = C-¹B-¹A-¹ and
your representation of N as a product from part (b).
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