Exercise 2.4.23

Answers

Matrix $A$ given by

A = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]

has $A^{2} = 0$ . Note: Any matrix

A = column times row = u v^{T}

will have

A^{2} = u v^{T} u v^{T} = 0

if $v^{T} u = 0$ . Thus,

A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}]

has

A^{2} = [\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

but $A^{3} = 0$ ; strictly triangular as in Problem 21.

mathchakra

2021-12-06 17:46