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Chapter 2: Q32E (page 54)

Find a n脳nmatrix A such thatAx鈫�=3x鈫�for allx鈫�inRn.

Short Answer

Expert verified

Matrix A will be 3I, whereI identity matrix of order is n脳n.

Step by step solution

01

Step by step Explanation: Step1: System of equation

We have given thatAx鈫�=3x鈫� forx鈫� all in Rn.

Which implies thatx鈫� isn脳1 matrix vector.

02

Linear Transformation

A transformation from Rm鈫�Rnis said to be linear if the following condition holds:

  1. Identity of Rmshould be mapped to identity of Rn.
  2. T(a+b)=T(a)+T(b)For all a,b鈭�Rm.
  3. T(ca)=cT(a)Where c is any scalar and a鈭�Rm.
03

Simplification

We have Ax鈫�=3x鈫�. We can write it as

Ax鈫�-3x鈫�=0(A-3)x鈫�=0

Here, we have taken the vector x is non-zero.

This implies that

A-3=0A=3I

Where Iis identity matrix of order n脳n.

Thus, matrix A will be 3I, where identity matrix of order is n脳n.

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Most popular questions from this chapter

If the matrix [abcdefghi] is invertible, then the matrix [abde] must be invertible as well.

Consider the circular face in the accompanying figure. For each of the matrices A in Exercises 24 through 30, draw a sketch showing the effect of the linear transformation T(x)鈫�=Ax鈫�on this face.

27.[100-1]

Let A=[110-312]in all parts of this problem.

(a) Find the scalar位 such that the matrixA-位濒2fails to be invertible. There are two solutions; choose one and use it in parts (b) and (c).

(b) For the 位you choose in part (a), find a non-zero vector x鈬赌such that

role="math" localid="1659697491583" (A-位濒2)x鈬赌=0鈬赌

(c) Note that the equation (A-位濒2)x鈬赌=0鈬赌can be written as.

A-位x鈬赌=0鈬赌orAx鈬赌=位x鈬赌Check that the equationAx鈬赌=位x鈬赌 holds for your位from part (a) and yourx鈬赌from part (b).

There exists an invertible 2 脳 2 matrix A such that A-1=[1111].

Give a geometric interpretation of the linear transformations defined by the matrices in Exercises16through 23 . Show the effect of these transformations on the letter L considered in Example 5. In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise 13.

22.[100-1]
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