Q26E Find the determinants of the lin... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q26E (page 289)

Find the determinants of the linear transformations in Exercises 17 through 28.
26. $T (M) = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] M + M [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}]$ from the space V of symmetric 2 脳 2 matrices to V

Short Answer

Expert verified

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

$d e t T = d e t B = - 16$

Step by step solution

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.

Given.

Given linear transformation,

$T (M) = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] M + M [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}]$

To find determinant.

For $M 鈭� V$ , we have

$M = [\begin{matrix} a & b \\ b & d \end{matrix}]$

So we compute

$T (M) = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] M + M [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] T (M) = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] [\begin{matrix} a & b \\ b & d \end{matrix}] + [\begin{matrix} a & b \\ b & d \end{matrix}] [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] T (M) = [\begin{matrix} 2 a + 4 b & 2 a + 4 b + 2 d \\ 2 a + 4 b + 2 d & 4 b + 6 d \end{matrix}]$

Since,

$B = \{[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]\}$ is a basis for $V$ , this means that the matrix of $T$ corresponding to $B$ is

$B = [\begin{matrix} 2 & 4 & 0 \\ 2 & 4 & 2 \\ 0 & 4 & 6 \end{matrix}]$

Therefore,

$d e t T = d e t B = - 16$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the determinants of the linear transformations in Exercises 17 through 28.
26. $T (M) = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] M + M [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}]$ from the space V of symmetric 2 脳 2 matrices to V

Short Answer

Step by step solution

Definition.

Given.

To find determinant.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

Find the determinants of the linear transformations in Exercises 17 through 28.26. T(M)=[1223]M+M[1223]from the space V of symmetric 2 脳 2 matrices to V

Short Answer

Step by step solution

Definition.

Given.

To find determinant.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find the determinants of the linear transformations in Exercises 17 through 28.
26. $T (M) = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] M + M [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}]$ from the space V of symmetric 2 脳 2 matrices to V