What happens if matrix $A^2$ has a zero column?

Question

Let me assume that $A$ is a square matrix and the matrix $A^2$ has a column of zeros. Is it possible for me to prove that A has a column of zeros. I know that the determinants are both zero, however it did not help me. I assumed that jth column is the zero row and had [i,j] term $(a_{i1}*a_{1j}+a_{i2}*a_{2j}+...+a_{ij}*a_{ij}+...+a_{in}*a_{nj}=0)$ equal to zero. I summed it for all $i$ in range 1 and $n$, and tried to prove it. But I was not able to do it. Are there anything I am missing?

Answer 1

Just because $A^2$ has a column of zeroes, doesn't mean $A$ does. For example, $$\begin{bmatrix}1&-1\\1&-1\end{bmatrix}^2=\begin{bmatrix}0&0\\0&0\end{bmatrix}.$$

Answer 2

To obtain other similar couterexamples let consider

$$\begin{bmatrix}a&b\\c&d\end{bmatrix}^2=\begin{bmatrix}a^2+bc&ab+cd\\ac+dc&cb+d^2\end{bmatrix}$$

and take for example

• $a^2+bc=0 \implies a=i, b=1, c=1$
• $ac+dc=0 \implies d=-i$

to obtain

$$\begin{bmatrix}i&1\\1&-i\end{bmatrix}^2=\begin{bmatrix}0&0\\0&0\end{bmatrix}$$