Question textThe linear transformation represented by a matrix 𝖠 transforms the unit square [math][0,1] imes [0,1] into a parallelogram with vertices [math: [0,0]]{\left[ 0 , 0 ight]}, [math: [3,0]]{\left[ 3 , 0 ight]}, [math: [2,5]]{\left[ 2 , 5 ight]}, and [math: [5,5]]{\left[ 5 , 5 ight]}. What is the magnitude of the determinant of 𝖠?[input] Check Question 40

Question

Question textThe linear transformation represented by a matrix 𝖠 transforms the unit square [math][0,1] 	imes [0,1] into a parallelogram with vertices [math: [0,0]]{\left[ 0 , 0 ight]}, [math: [3,0]]{\left[ 3 , 0 ight]}, [math: [2,5]]{\left[ 2 , 5 ight]}, and [math: [5,5]]{\left[ 5 , 5 ight]}. What is the magnitude of the determinant of 𝖠?[input] Check Question 40

BlackTom AI · Accepted Answer

The problem describes a linear transformation A that sends the unit square with corners (0,0), (1,0), (0,1), (1,1) to a parallelogram whose vertices include (0,0), (3,0), (2,5), and (5,5). In such a mapping, the two edge vectors from the origin become the column vectors of the transformation’s matrix: the vector from (0,0) to (3,0) is (3,0), and the vector from (0,0) to (2,5) is (2,5). Therefore, a natural choice for the 2x2 matrix A (with columns being these edge vectors) is:

A = [ [3, 2], [0, 5] ]

The determinant of a 2x2 matrix det([ [a, b], [c, d] ]) equals ad − bc. Substituting a = 3, b = 2, c = 0, d = 5 gives det(A) = 3·5 − 0·2 = 15. Since the problem asks for the magnitude of the determinant, and the determinant here is already nonnegative, the magnitude is |15| = 15.

Intuitively, the determinant of the matrix formed by the two edge vectors gives the oriented area scaling from the unit square to the parallelogram; the area of the parallelogram equals 15, confirming the determinant magnitude of 15.

Note on approach: if one chose the other valid representation of A with rows rather than columns as edge vectors, the computed determinant would be the same in magnitude (possibly with a sign change depending on orientation), so taking magnitude ensures a nonnegative value that matches the parallelogram’s area.

In summary, using the edge vectors (3,0) and (2,5) as columns yields det(A) = 15, consistent with the parallelogram area.

ENG1005 - MUM S2 2025 [FINAL REVISION] Quizzes

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