# Matrix operations that preserve the nullspace of a matrix?

Context: Let $A$ be matrix of nullity $> 0.$ I want to compute the nullspace of $A.$ But in certain cases, it's easier to transform $A$ into another matrix, then compute the nullspace of the resulting matrix, given that the transformation preserves the nullspace. For example: $\mbox{nullspace}(A) = \mbox{nullspace}(A^t A).$

My Question: In addition to multiplying $A$ by its transpose, what are the other matrix operations that preserve the nullspace of $A$?

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Have you heard of Gaussian Elimination? –  Dan Donnelly Apr 7 '11 at 21:35
$A \to SA$ where $S$ is an invertible matrix, or more generally a matrix such that the intersection of the nullspace of $S$ and the column space of $A$ is $\{0\}$ (this includes the case $S = A^t$, as well as the elementary row operations).