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Did you know that the determinant of a matrix has some fascinating properties? One such property is that if you cyclically permute the rows of a matrix, the determinant remains unchanged. So, for the matrix you provided, moving the first row to the bottom (and all others up) keeps the determinant the same. Therefore, .

Now, let’s highlight a common mistake when calculating determinants. Many forget the sign changes incurred during row swaps. Remember that if you swap any two rows, the determinant switches sign! So always keep track of those swaps if you decide to manipulate rows while calculating. Consistency is key!