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.

类似问题

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:

What was the world's first widely adopted biodiversity policy?

The aim of the Paris Agreement is to ensure that the earth’s preindustrial temperature is not exceeded by more than º Celsius.

The IPCC's mission is to Blank ______.

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单