Science:Math Exam Resources/Courses/MATH152/April 2012/Question 06 (c)

MATH152 April 2012

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[hide] Question 06 (c)
This question requires little or no calculations. Let $A=M_{{\text{Rot}}\,60^{\circ }}$ be the matrix that rotates a 2D vector counter clockwise by $60^{\circ }$ , $B=M_{{\text{Ref}}\,60^{\circ }}$ be the matrix that reflects in the line $y={\sqrt {3}}x$ , and $C=M_{{\text{Proj}}\,45^{\circ }}$ be the projection in the line y=x. Which three of the following six statements are true? ${\begin{aligned}&{\textrm {(1)}}&A^{6}&=I\\&{\textrm {(2)}}&A^{-1}&=A\\&{\textrm {(3)}}&B^{3}&=I\\&{\textrm {(4)}}&B^{-1}&=B\\&{\textrm {(5)}}&C^{-1}&=C\\&{\textrm {(6)}}&C^{2}&=C\end{aligned}}$

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[show] Hint
Consider what happens by repeatedly applying the matrices to themselves. For example, if we apply the rotate by 60 degrees matrix twice, how much does the resulting matrix rotate in total? Similarly, if we reflect a vector twice about a line, where does it end up? What happens if we reflect a vector to the same line twice? Are all these operations invertible?

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[show] Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. ${\color {blue}{\begin{aligned}&{\textrm {(1)}}&A^{6}&=I&{\textrm {(True)}}\\&{\textrm {(2)}}&A^{-1}&=A&{\textrm {(False)}}\\&{\textrm {(3)}}&B^{3}&=I&{\textrm {(False)}}\\&{\textrm {(4)}}&B^{-1}&=B&{\textrm {(True)}}\\&{\textrm {(5)}}&C^{-1}&=C&{\textrm {(False)}}\\&{\textrm {(6)}}&C^{2}&=C&{\textrm {(True)}}\end{aligned}}}$ The quick solution to this problem involves using some understanding about how applying the matrices above to any arbitrary vector changes the orientation of that vector. The reasoning is explained in the following points: (1): Applying the matrix $\displaystyle A$ to a vector once will rotate the vector counterclockwise by 60 degrees. Applying the matrix $\displaystyle A$ to a vector twice (i.e. $A^{2}$ ) will rotate the vector counterclockwise by 120 degrees... Applying it to a vector 6 times (i.e. $A^{6}$ ) will rotate the vector counterclockwise by 360 degrees... giving us the original vector back, just as applying the identity matrix $I$ would do. Hence (1) is true. (2): If $v=Au$ , then v the 60 degree counterclockwise rotated version of u. Also if $v=Au$ , then $u=A^{-1}v$ , which tells us that $A^{-1}$ is the matrix corresponding to 60 degree clockwise rotation. So $A,A^{-1}$ are not the same matrix operator and (2) is false. (3): Since the matrix $B$ will reflect a vector across the line $y={\sqrt {3}}x$ , applying $B$ twice to a vector will reflect the vector across the same line twice, resulting in the original vector. Hence applying the matrix $B$ three times to a vector will reflect it across the line. This is not equivalent to multiplying the vector by $I$ and so (3) is false. (4): If $B^{-1}=B$ , then $I=B^{2}$ which, as we mentioned in (3), is true since reflecting a vector across the line twice gives the original vector. Therefore, (4) is true. (5): Projections are non-invertible operations (since you cannot uniquely determine the original vector from its projection) and since $C$ is a projection, $C^{-1}$ does not even exist. So (5) is false. (6): Since $C$ is a projection, it satisfies (6) by definition and so (6) is true.