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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: Q7E (page 107)

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

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Consider two matrices

Let $A = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$ and $B = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}]$

02

Find the product of the matrices

Find the product AB and BA.

$A B = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] = [\begin{matrix} 3 & 4 \\ 1 & 2 \end{matrix}] B A = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}] = [\begin{matrix} 2 & 1 \\ 4 & 3 \end{matrix}]$

Hence, $A B 鈮� B A$ .

So, the given statement is false.

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Most popular questions from this chapter

For which values of the constants b and c is the following matrix invertible?

$[\begin{matrix} 0 & 1 & b \\ - 1 & 0 & - c \\ - b & - c & 0 \end{matrix}]$

Find a $n 脳 n$ matrix A such that $A \overset{鈫�}{x} = 3 \overset{鈫�}{x}$ for all $\overset{鈫�}{x}$ in $R^{n}$ .

TRUE OR FALSE?

Matrix $[\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}]$ is invertible.

Consider an invertible $n 脳 n$ matrix Aand $a n n 脳 n$ matrix B.A certain sequence of elementary row operations transforms Ainto $I_{n}$ .

a. What do you get when you apply the same row operations in the same order to the matrix AB?

b. What do you get when you apply the same row operations to $I_{n}$ ?.

Find the matrices of the transformations Tand Ldefined in Exercise 80.

See all solutions

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