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# Application of matrices

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After applying $$\mathbf M = \begin{pmatrix} -1 & 0 \\ -\sqrt 2 & 3 \end{pmatrix}$$to the circle of radius 3 centered at (2,0), what is the area of the resulting region?

Here's what I've done so far: I found two points - (2,0) and (-1,0), which I turned into vectors of $$\ \begin{pmatrix} 2\\ 0 \end{pmatrix}$$and $$\ \begin{pmatrix} -1\\ 0 \end{pmatrix}$$ . I then multiplied these by the original matrix, to get $$\ \begin{pmatrix} -2\\ -2\sqrt2 \end{pmatrix}$$ and $$\ \begin{pmatrix} 1\\ \sqrt2 \end{pmatrix}$$ . I then calculated the distance between these two using the distance formula, which I got to equal $$\sqrt11$$. I then plugged in $$\sqrt11$$ to the circle area formula, ending up with a circle with area $$​​11\pi$$. However, this answer is incorrect. Any tips on how I should solve this? Thanks!

Mar 20, 2021

#1
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I got it, nevermind! I calculated the distance incorrectly. The answer was $$27\pi$$.

Mar 20, 2021
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distance  between   ( -2 , -2sqrt (2) )  and   ( 1 , sqrt (2) ) =

sqrt  [ ( 1 - -2)^2  +  ( sqrt 2 - - 2 sqrt 2)^2  ]  =

sqrt [ 3^2  + ( 3sqrt (2)) ^2 ] =

sqrt  [ 9 +  18 ] = sqrt (27)

Area =  pi  (sqrt (27) )^2  = 27 pi   Mar 20, 2021