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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: Q18E (page 108)

TRUE OR FALSE?
Matrix $(\begin{matrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix})$ is invertible.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Step 1: Invertible matrix

The matrix is said to be invertible if the reduced matrix has nonzero diagonal elements.

02

Compute the reduced form of the matrix

Consider the matrix.

$(\begin{matrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix})$

The reduced form of the matrix is,

role="math" localid="1659351901635" $(\begin{matrix} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix}) = (\begin{matrix} 1 & 1 & 1 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}) = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix})$

As the diagonal elements are nonzero, thus, the matrix is invertible.

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Most popular questions from this chapter

If $A^{2}$ is invertible, then matrix A itself must be invertible.

Write all possible forms of elementary $2 脳 2 E_{5} = [\begin{matrix} 1 & 0 \\ k & 1 \end{matrix}]$ matrices E. In each case, describe the transformation $\overset{鈫�}{y} = E \overset{鈫�}{x}$ geometrically.

Consider a block matrix $A = [\begin{matrix} A_{11} & 0 \\ 0 & A_{22}^{} \end{matrix}]$ , where role="math" localid="1660371822794" $A_{11} and A_{12}$ are square matrices. For which choices of $A_{11} and A_{22}$ is A invertible? In these cases, what is ?

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?

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

TRUE OR FALSE?

There is a transition matrix A such that $\underset{m 鈫� 鈭�}{l i m} A^{m}$ fails to exist.

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