# Square root of a 0 matrix

by Bipolarity
Tags: matrix, root, square
 P: 783 At first I thought that there is no square matrix whose square is the 0 matrix. But I found a counterexample to this. My counterexample is: $$\left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right)$$ However it appears that my counterexample has a 0 row. I'm curious, must a square root of the 0 matrix necessarily have at least one 0 row (or 0 column)? BiP
 Quote by Bipolarity At first I thought that there is no square matrix whose square is the 0 matrix. But I found a counterexample to this. My counterexample is: $$\left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right)$$ However it appears that my counterexample has a 0 row. I'm curious, must a square root of the 0 matrix necessarily have at least one 0 row (or 0 column)? BiP
 HW Helper P: 6,187 I suspect you intended the following matrix? $$\begin{bmatrix}0 & 1 \\ 0 & 0 \end{bmatrix}$$ Square it and you get the zero matrix. The same holds for $$\begin{bmatrix}1 & 1 \\ -1 & -1 \end{bmatrix}$$