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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: Q65 E (page 73)

Find all upper triangular $2 脳 2$ matricesXsuch that X²is the zero matrix.

Short Answer

Expert verified

Thus, the matrix X such that $X^{2}$ is the zero matrix is $[\begin{matrix} 0 & 0 \\ 0 & t \end{matrix}]$ , where t is any arbitrary.

Step by step solution

01

Assuming the matrix

Let the upper triangular matrix of $2 脳 2 i s [\begin{matrix} a & c \\ 0 & b \end{matrix}]$

Now,

$X = [\begin{matrix} a & c \\ 0 & b \end{matrix}] X^{2} = [\begin{matrix} a & c \\ 0 & b \end{matrix}] [\begin{matrix} a & c \\ 0 & b \end{matrix}] = [\begin{matrix} a^{2} & a c + c b \\ 0 & b^{2} \end{matrix}]$

02

Solving the Matrix

Since we have to put

X^{2} = [\begin{matrix} a^{2} & a c + c b \\ 0 & b^{2} \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]

On equating the matrix values to 0, we get

$a^{2} = 0, b^{2} = 0, a c + c b = 0 鈬� a = 0, b = 0 c (a + b) = 0$

Thus, the matrix X such that $X^{2}$ is the zero matrix is $[\begin{matrix} 0 & 0 \\ 0 & t \end{matrix}]$ . Where t is any arbitrary.

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Most popular questions from this chapter

TRUE OR FALSE?

$A = [\begin{matrix} 1 & 1 / 2 \\ 0 & 1 / 2 \end{matrix}]$ is a regular transition matrix.

Consider a linear transformation Tfrom $R^{2}$ to $R^{2}$ . Suppose that $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ are two arbitrary vectors in $R^{2}$ and that $\overset{鈫�}{x}$ is a third vector whose endpoint is on the line segment connecting the endpoints of $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ . Is the endpoint of the vector $T (\overset{鈫�}{x)}$ necessarily on the line segment connecting the endpoints of $T (\overset{鈫�}{v)}$ and $T (\overset{鈫�}{w)}$ ? Justify your answer.

Use the concept of a linear transformation in terms of the formula $y = A \overset{鈬赌}{x}$ , and interpret simple linear transformations geometrically. Find the inverse of a linear transformation from localid="1659964769815" ${鈩�}^{2} t o {鈩�}^{2}$ to (if it exists). Find the matrix of a linear transformation column by column.

Consider the transformations from ${鈩�}^{3} t o {鈩�}^{3}$ defined in Exercises 1 through 3. Which of these transformations are linear?

$y_{1} = 2 x_{2} y_{2} = x_{2} + 2 y_{3} = 2 x_{2}$

There exists a positive integer n such that ${[\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}]}^{n} = l_{2}$ .

Some parking meters in downtown Geneva, Switzerland, accept $2$ Franc and $5$ Franc coins.

a. A parking officer collects $51$ coins worth $144$ Francs. How many coins are there of each kind?

b. Find the matrix $A$ that transforms the vector

$[\begin{matrix} number of 2 Franc coins \\ number of5Franc coins \end{matrix}]$

into the vector

$[\begin{matrix} total value of coins \\ totalnumberof coins \end{matrix}]$

c. Is the matrix $A$ in part (b) invertible? If so, find the inverse (use Exercise 13). Use the result to check your answer in part (a).

See all solutions

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