2x2 Matrix with no zero entries where $A^k=0$ - Nilpotence?

Question

Find an example of a $2x2$ matrix $A$ that has no zero entries but is such that $A^K=0$ for some positive integer k.

Here is my thinking: When $k=1, A=0$, but this contradicts that the matrix has no zero entries, so no such matrix exists. Then I started reading about nilpotence and I got very confused. Can someone explain this to me? What am I missing?

Why isn't it that no such matrix exists, considering k=1?

Answer 1

$\pmatrix{1&-1\cr 1&-1}$ is an example

Answer 2

Let the matrix $A$ have elements

$a$ $b$

$c$ $d$

You get $A^2=0$ if the characteristic polynomial

$\det(A-\lambda I)=(\lambda^2-(a+d)\lambda+(ad-bc))=0$

Then $d=-a$ and $bc=ad=-a^2$.

Try out various values of $a$ and $b$, work out $c$ and $d$ from the equations just derived, and watch what happens.

Answer 3

If $k=1$ and $A^k=0$ then $A^1=A=0$, i.e., $A$ is the zero matrix, with zero entries, so there is no solution to $A^k=0$ with $k=1$ and $A$ having non-zero entries.

Answer 4

Let

$$A=\left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)$$

Then write down the matrix equation

$$A.A = 0$$

This gives you four equations (one for each position in the Matrix equation)

Solving these equations for $a$ and $b$ gives you the solution

$$A_1 = \left( \begin{array}{cc} a & b \\ -\frac{a^2}{b} & -a \\ \end{array} \right)$$

valid for any $a$ but $b\ne 0$.

If $b=0$ the solution is

$$A_2 = \left( \begin{array}{cc} 0 & 0 \\ c & 0 \\ \end{array} \right)$$

for any $c$.

By symmetry, if $c=0$ we have for any $b$

$$A_3 = \left( \begin{array}{cc} 0 & b \\ 0 & 0 \\ \end{array} \right)$$

Notice that $A_3 = A_1(a=0)$.