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Q8E

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Linear Algebra With Applications
Found in: Page 57
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

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Short Answer

Question 8: Find the inverse of linear transformation

y1=x1+7x2y2=3x1+20x2

Answer:

Inverse of the transformation isT-1(x1,x2)=(-20x1+7x2,3x1-x2) .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step1: System of equations

We have given a linear transformation with

y1=x1+7x2y2=3x1+20x2

Step2: Linear Transformation

Let T be linear transformation from R2→R2.

Then, according to given equations.

Tx1x2=x1+7x23x1+20x2

Step3: Matrix for linear transformation

The matrix representation of T by 2x2 matrix.

.A=17320

Inverse of matrix is calculated as

First of all we will find the determinant of matrix.

A=20-21=-1adjA=20-7-31

Inverse of matrix= .A-1=-2073-1

Hence, inverse of the transformation will be

T-1(x1,x2)=(-20x1+7x2,3x1-x2)

Hence, inverse of the transformation isdata-custom-editor="chemistry" T-1(x1,x2)=(-20x1+7x2,3x1-x2)

Most popular questions for Math Textbooks

Consider an n × p matrix A, a p × m matrix B, and a scalar k. Show that (k A)B = A (k B) = k(AB)

In this exercise we will verify part (b) of Theorem 2.3.11 in the special case when A is the transition matrix [0.40.30.60.7] and x¯is the distribution vector[10] . [We will not be using parts (a) and (c) of Theorem 2.3.11]. The general proof of Theorem 2.3.11 runs along similar lines, as we will see in Chapter 7.

  1. Compute A [12] and A [1-1] . Write A [1-1]as a scalar multiple of the vector [1-1] .
  2. Write the distribution vectorx→ = [10] as a linear combination of the vectors [12] and [1-1]
  3. Use your answers in part (a) and (b) to write Ax→ as a linear combination of the vectors [12] and [1-1] . More generally, write Amx→as a linear combination of vectors [12] and [1-1] , for any positive integer m. See Exercise 81.
  4. In your equation in part (c), let got to infinity to find limm→∞Amx→. Verify that your answer is the equilibrium distribution for A.

TRUE OR FALSE?

There exists an invertible matrix S such that S-1(0-100) is a diagonal matrix.

There exists a positive integer n such that [0-110]n=l2.

TRUE OR FALSE?

There exists an invertible 2×2 matrix A such that A2=(100-1) .

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