Q28E There exists a nonzero upper tri... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q28E (page 108)

There exists a nonzero upper triangular 2 脳 2 matrix A such that $A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ .

Short Answer

Expert verified

The statement is false.

Step by step solution

Proving statement by example

The claim is that there exists an upper triangular 2x2 matrix A such that $A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ .

First, let A= $[\begin{matrix} a & b \\ 0 & c \end{matrix}]$ from here we see that:

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

Conclusion

Now if $A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ this means that: $[\begin{matrix} a^{2} & a b + b c \\ 0 & c^{2} \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

$鈬� a = 0$ but c = 0 under these conditions, A is no longer an upper triangular matrix and thus the statement is false.

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91影视

There exists a nonzero upper triangular 2 脳 2 matrix A such that $A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ .

Short Answer

Step by step solution

Proving statement by example

Conclusion

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91影视

There exists a nonzero upper triangular 2 脳 2 matrix A such that A2=[0000].

Short Answer

Step by step solution

Proving statement by example

Conclusion

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There exists a nonzero upper triangular 2 脳 2 matrix A such that $A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ .