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Problem 5 Easy Difficulty

If any row (or any column) of a determinant is multiplied by a nonzero number k, the value of the determinant remains unchanged.




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Video Transcript

you're given the statement. If any row or any column of a determinant is multiplied by a non zero number such as K, the value of the determinant remains unchanged on our goal. Here is to figure out if this is a true or false statement. So if we look at an example of a determinant for The Matrix, whose first row is four in three second rows eight and two, this is just a random example to show you that we can either prove or disprove this statement. So recall that we use the diagonals to determine the determinant and this would result in four times two minus eight times three. This gives us eight minus 24 resulting in it determinant of negative 16. So keep that in mind for just a minute. Now. If we take any non zero values such as K and multiply it by our determinant, I say we do it in any row. To start, we would have four K three k and then eight and two would remain. We follow the same process to determine the determinant by using the diagonals, and we again find that we have four K times two minus eight times three k. This results in eight K minus 24 k. When simplifying here, we would now have negative 16 k. The determinant of K d is negative. 16 k Therefore, the determinant is changed by the scale factor of K. The determinant remains unchanged is a false statement.