Question about Multiplying Matrices

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Row reducing two matrices before multiplication does not yield a row-equivalent result. Specifically, when multiplying two 2x2 matrices, row reduction can lead to a zero in the bottom left entry of the product, which is not guaranteed in the original matrices. This demonstrates that the product of two upper triangular matrices remains upper triangular, but not all 2x2 matrix products share this property. Therefore, the operation of row reduction alters the fundamental characteristics of the matrices involved.

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If I reduce two matrices using row operations before multiplying them together, will I still get a row-equivalent answer to the result I would of gotten if I hadn't reduced them?

Thanks for any input!
 
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In short, no. If we look at multiplying two 2x2 matrices, row reducing them first would leave 0 in the bottom left entry in each one, and their product will have a 0 in that entry as well. (This would be an example of two upper triangular matrices being multiplied, and their product will always be upper triangular too.) But not every product of 2x2 matrices will have a 0 in the bottom left corner, so in general, row reducing them first won't give you a row-equivalent answer.
 

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