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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q24E (page 289)

Find the determinants of the linear transformations in Exercises 17 through 28.
24. $T (z) = (2 + 3 i) z f r o m 鈩� t o 鈩�$

Short Answer

Expert verified

Therefore, the determinant of the linear transformations is given by,

$d e t T = d e t B = 13$

Step by step solution

01

Definition.

A determinant is a unique number associatedwith a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

02

Given.

Given linear transformation,

$T (z) = (2 + 3 i) z f r o m 鈩� t o 鈩�$

03

To find determinant.

Consider the linear transformation $T : 鈩� 鈫� 鈩�$ defined as. $T (z) = (2 + 3 i) z$

Consider the basis $B = \{1, i\}$ for $鈩�$ .
We have that

and

$T (1) = (2 + 3 i) 1 = 2 + 3 i and T (i) = (2 + 3 i) i = 2 i + 3 i^{2} = - 3 + 2 i$

The matrix of corresponding to B is

$B = [\begin{matrix} 2 & 3 \\ - 3 & 2 \end{matrix}]$

Observe that

$\det B = 2.2 - (- 3) . 3 \det B = 4 - (- 9) \det B = 13$

Therefore,

det T = det B = 13

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Most popular questions from this chapter

a. Find a noninvertible $2 脳 2$ matrix whose entries are four distinct prime numbers, or explain why no such matrix exists.

b. Find a noninvertible $3 脳 3$ matrix whose entries are nine distinct prime numbers, or explain why no such matrix exists.

Find the determinants of the linear transformations in Exercises 17 through 28.

25. $T (M) = [\begin{matrix} 2 & 3 \\ 0 & 4 \end{matrix}]$ M from the space V of upper triangular matrices to V

Consider the quadrilateral in the accompanying figure, with vertices $P_{i} = (x_{i}, y_{i})$ , for $i = 1, 2, 3, 4$ . Show that the area of this quadrilateral is

$\frac{1}{2} (d e t [\begin{matrix} x_{1} & x_{2} \\ y_{1} & y_{2} \end{matrix}] + d e t [\begin{matrix} x_{2} & x_{3} \\ y_{2} & y_{3} \end{matrix}] + d e t [\begin{matrix} x_{3} & x_{4} \\ y_{3} & y_{4} \end{matrix}] + d e t [\begin{matrix} x_{4} & x_{1} \\ y_{4} & y_{1} \end{matrix}])$ .

If all the entries of a $7 脳 7$ matrix A are 7, then $d e t A$ must be $7^{7}$ .

In an economics text $^{11}$ we find the following system:

$[\begin{matrix} - R_{1} & R_{1} & - (1 - 伪) \\ 伪 & 1 - 伪 & - {(1 - 伪)}^{2} \\ R_{2} & - R_{2} & \frac{- {(1 - 伪)}^{2}}{伪} \end{matrix}] [\begin{matrix} d x_{1} \\ d y_{1} \\ d p \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ - R_{2} d e_{2} \end{matrix}]$ .

Solve for $d x_{1}, d y_{1}$ , and dp. In your answer, you may refer to the determinant of the coefficient matrix as D. (You need not compute D.) The quantities $R_{1}, R_{2}$ and D are positive, and ais between zero and one. If $d e_{2}$ is positive, what can you say about the signs of $d y_{1}$ and dp?

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