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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: Q15E (page 108)

RUE OR FALSE?
The formula $\det (2 A) = 2 \det (A)$ holds for all $2 脳 2$ matrices A.

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Determinant of a matrix

The determinant of $2 脳 2$ matrix is, $d e t (\begin{matrix} a & b \\ c & d \end{matrix}) = a d - b c$ .

02

Compute the determinant of the matrix

Consider the matrix.

$A = (\begin{matrix} a & b \\ c & d \end{matrix})$

The determinant of the matrix is,

$2 d e t (A) = 2 d e t (\begin{matrix} a & b \\ c & d \end{matrix}) = 2 a d - b c$

Consider the matrix.

$2 A = (\begin{matrix} 2 a & 2 b \\ 2 c & 2 d \end{matrix})$

The determinant of the matrix is,

$d e t (2 A) = d e t (\begin{matrix} 2 a & 2 b \\ 2 c & 2 d \end{matrix}) = 4 (a d - b c)$

Therefore, $\det (2 A) 鈮� 2 \det (A)$ .

Hence, the given statement is false.

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Most popular questions from this chapter

TRUE OR FALSE?

If the linear system $A^{2} \overset{鈫�}{x} = \overset{鈫�}{b}$ is consistent, then the systemrole="math" localid="1659352762883" $A \overset{鈫�}{x} = \overset{鈫�}{b}$ must be consistent as well.

Use the formula derived in Exercise $2.1.13$ to find the inverse of the rotation matrix

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Use part (c) of Theorem 2.3.11 to prove part (b): If Ais a regular transition matrix of size $n 脳 n$ with equilibrium distribution ${\overset{鈫�}{x}}_{e q u}$ , and if $\overset{鈫�}{x}$ is any distribution vector in ${鈩�}^{n}$ , then $\underset{m 鈫� 鈭�}{l i m} (A^{m} \overset{鈫�}{x}) = {\overset{鈫�}{x}}_{e q u}$ .

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