When Elementary Row Operations (EROs) are taught in an introductory setting (for instance, a one-semester multi-department undergrad course), they are often presented as steps in "mental algorithms" to find the inverse of a matrix, or to solve systems of equations. "How to use EROs" is often far more emphasized than "What EROs are". For those who share this experience, row operations remain mysterious and intriguing.
How does one understand what an Elementary Row Operation is? Can EROs be described in such a way that makes their usefulness more apparent?
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