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

View Explanation

Log in for full answers

Similar Questions

A is a ( 2 × 2 ) matrix and the inverse of 1 3 𝐴 is the identity matrix. What is the determinant of A?

If matrix \(A = \left[ {\begin{array}{*{20}{c}}5&6\\5&{ - 4}\end{array}} \right]\), then \(\det \left( A \right) = \)

The matrix [math: [234x+1]] \left[\begin{array}{cc} 2&3 \\ 4&x+1 \end{array}\right] will NOT have an inverse where:

The matrix [math: [234x+1]] \left[\begin{array}{cc} 2&3 \\ 4&x+1 \end{array}\right] will NOT have an inverse where:

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler