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Let R be a region with area 3. The matrix \(\begin{bmatrix} 2 && 1 \\ 3 && -7 \end{bmatrix}\) is applied to R, resulting in the region R'. FInd the area of region R'.\(\)

 May 31, 2022
 #1
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The area of R' is 12.

 May 31, 2022
 #2
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Thanks for posting the "solution" without an explanation! But it would be nice if you showed me how you solved it :)

MobiusLoops  May 31, 2022
 #3
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Hello MobiusLoops,

When a matrix is applied on a region, the determinant of the matrix is the scale factor of the enlargement that occurs.

That is, the determinant of the matrix: \(\begin{bmatrix} 2 && 1 \\ 3 && -7 \end{bmatrix}\) is: \(2(-7)-1(3)=-17\)  

(In general, the determinant of: \(\begin{bmatrix} a && b \\ c && d \end{bmatrix} \) is \(ad-bc\) )

So, \(3(-17)=-51\), since It is an area, then we take the absolute value of this number, giving the answer to be 51, which is the area of R'.

(The determinant of 3x3 matrix, will be the "Volume scale factor" and the determinant of 2x2 matrix will be "Area scale factor (as above).")

 Jun 1, 2022
 #4
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Thanks for the explanation!

MobiusLoops  Jun 1, 2022

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