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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 3: Q35E (page 164)

If AB = 0 for two 2 x 2 matrices A and B, then BA must be zero matrix as well.

Short Answer

Expert verified

The above statement is false.

If AB = 0 for two 2 x 2 matrices A and B, then BA may not be a zero matrix.

Step by step solution

01

Taking an example

Let $A = [\begin{matrix} 0 & - 1 \\ 0 & 2 \end{matrix}] andB= [\begin{matrix} 3 & 5 \\ 0 & 0 \end{matrix}]$ are two 2x2 matrices such that

$A B = [\begin{matrix} 0 & - 1 \\ 0 & 2 \end{matrix}] [\begin{matrix} 3 & 5 \\ 0 & 0 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

$鈬� A B = 0$

Now,

$B A = [\begin{matrix} 3 & 5 \\ 0 & 0 \end{matrix}] [\begin{matrix} 0 & - 1 \\ 0 & 2 \end{matrix}] = [\begin{matrix} 0 & 7 \\ 0 & 0 \end{matrix}]$

$鈬� B A 鈮� 0$

02

Final Answer

Since we have find two 2 X 2 matrices $A = [\begin{matrix} 0 & - 1 \\ 0 & 2 \end{matrix}] andB= [\begin{matrix} 3 & 5 \\ 0 & 0 \end{matrix}]$ such that

$A B = 0 but BA 鈮� 0$ , hence the given statement is false.

Thus, if AB = 0 for two 2 x 2 matrices A and B, then BA may not be a zero matrix.

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Most popular questions from this chapter

Two subspacesV andW of ${鈩�}^{n}$ are called complements if any vector $\overset{鈫�}{x}$ in ${鈩�}^{n}$ can be expressed uniquely as $\overset{鈫�}{x} = \overset{鈫�}{v} + \overset{鈫�}{w}$ , where $\overset{鈫�}{v}$ in V and $\overset{鈫�}{w}$ is in W. Show thatV andW are complements if (only if) can $V 鈭� W = {\overset{鈫�}{0}}$ and $\dim (V) + \dim (W) = n$ .

In Problem 46 through 55, Find all the cubics through the given points. You may use the results from Exercises 44 and 45 throughout. If there is a unique cubic, make a rough sketch of it. If there are infinitely many cubics, sketch two of them.

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