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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 2: Q36E (page 108)

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

Short Answer

Expert verified

The statement is false.

Step by step solution

Explaining

Consider a matrix A satisfying the condition,

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

Consider a matrix,

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

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.

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Most popular questions from this chapter

Question 7: Suppose $\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{2}}, \overset{鈫�}{v_{3}} 鈰� \overset{鈫�}{v_{m}}$ are arbitrary vectors in Consider the linear transformation from togiven by

$T [x_{1} x_{2} 鈰� x_{m}] = [x_{1} \overset{鈫�}{v_{1}}, x_{2} \overset{鈫�}{v_{2}}, x_{3} \overset{鈫�}{v_{3}} 鈰� x_{m} \overset{鈫�}{v_{m}}]$

TRUE OR FALSE?

There exists a matrix A such that $[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] A [\begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix}] = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ .

If the matrix $[\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}]$ is invertible, then the matrix $[\begin{matrix} a & b \\ d & e \end{matrix}]$ must be invertible as well.

In the financial pages of a newspaper, one can sometimes find a table (or matrix) listing the exchange rates between currencies. In this exercise we will consider a miniature version of such a table, involving only the Canadian dollar $(C \()$ and the South African Rand $(Z A R)$ . Consider the matrix

role="math" localid="1659786495324" $\begin{array}{rcl} \begin{matrix} C\) & ZAR \end{matrix} \\ A & = & [\begin{matrix} 1 & 1 / 8 \\ 8 & 1 \end{matrix}] \begin{matrix} C \( \\ ZAR \end{matrix} \end{array}$

representing the fact thatrole="math" localid="1659786520551" $C \) 1$ is worth role="math" localid="1659786525050" $Z A R 8$ (as of September 2012).

a. After a trip you have $C $ 100$ and $Z A R 1, 600$ in your pocket. We represent these two values in the vector $\overset{鈫�}{x} = [\begin{matrix} 100 \\ 1, 600 \end{matrix}]$ . Compute $A \overset{鈫�}{x}$ . What is the practical significance of the two components of the vector $A \overset{鈫�}{x}$ ?

b. Verify that matrix $A$ fails to be invertible. For which vectors $\overset{鈫�}{b}$ is the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ consistent? What is the practical significance of your answer? If the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ is consistent, how many solutions $\overset{鈫�}{x}$ are there? Again, what is the practical significance of the answer?

The formula $A B = B A$ holds for all $n 脳 n$ matrices A and B .

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ifA2=I2 then matrix A must be eitherI2or-I2 .

Short Answer

Step by step solution

Explaining

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