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can a 3 by 4 matrix be multiplied by a 4 by 1 matrix

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The general rule is that the number of columns of the first matrix must be equal to the number of rows of the second. To answer your question specifically, yes you can multiply a 3x4 with a 4x1 but you can't multiply a 4x1 with a 3x4. If you try it you'll see why, it's because you run out of entries to perform the operation on

Yes  3x4 4x1  notice that the columns in the first matrix matches the rows of the second...if they match, then you can multiply....the answer will be a 3x1 matrix