While evaluating the rank of a matrix is it permissible to apply row and column operations simultaneously on a single matrix? Most of the books that I discussed use either row or column operation (but not both) to evaluate the rank. May I apply both on a single matrix to evaluate rank?

  • $\begingroup$ One may always apply a sequence of row operations and column operations of a $n \times n$ matrix $A$ to arrive at $I_r \oplus 0_t$ where $r$ is the rank of the matrix and $t$ is the dimension of its kernel. For a more in-depth explanation, see this answer. $\endgroup$ – walkar Oct 9 '15 at 13:42

The rank of a matrix is invariant under application of elementary row and column operations.

So the answer is yes: Any mixture of row and column operations may be applied to determine the rank.

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