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Q36E

Expert-verified

Found in: Page 108

Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

if $A^{2} = I_{2}$ then matrix A must be either $I_{2} o r - I_{2}$ .

The statement is false.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Explaining

Consider a matrix A satisfying the condition,

$A^{2} = I_{2}$

Consider a matrix,

$\begin{array}{l} A = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] \\ \neq I_{2} \end{array}$

Step 2: Result

Calculate $A^{2}$

$A^{2} = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = I_{2}$

Thus the given statement is false.

Most popular questions for Math Textbooks

Consider the transformation T from $R^{2}$ to role="math" localid="1659714471562" $R^{2}$ that rotates any vector $\vec{x}$ through a given angle θ in the counterclockwise direction. Compare this with Exercise 33. You are told that T is linear. Find the matrix of T in terms of θ.

Give a geometric interpretation of the linear transformations defined by the matrices in Exercises 16 through 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 .

23.role="math" localid="1659695358882" $[\begin{matrix} 0 & 2 \\ - 2 & 0 \end{matrix}]$

Which of the functions f from R to R in Exercises 21 through 24 are invertible? 22 $f (x) = 2^{x}$ .

TRUE OR FALSE?

There exists an invertible $2 \times 2$ matrix A such that $A^{- 1} = - A$ .

a. Show that if an invertible $n \times n$ matrix A admits an LU-factorization, then it admits an LDU factorization. See Exercise 90 d.

b. Show that if an invertible matrix A admits an L DU-factorization, then this factorization is unique. Hint: Suppose that $A = L_{1} D_{1} U_{1} = L_{2} D_{2} U_{2}$ , then $U_{2} U^{- 1} = D_{2}^{- 1} L_{2}^{- 1} L_{1} D_{1}$ is diagonal (why?). Conclude that $U_{2} = U_{1}$ .

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Short Answer

if $A^{2} = I_{2}$ then matrix A must be either $I_{2} o r - I_{2}$ .

Step by Step Solution

Step 1: Explaining

Step 2: Result

Most popular questions for Math Textbooks

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Linear Algebra With Applications

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if A2=I2 then matrix A must be either I2 or -I2 .

Step by Step Solution

Step 1: Explaining

Step 2: Result

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if $A^{2} = I_{2}$ then matrix A must be either $I_{2} o r - I_{2}$ .