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Simplifying Determinants - Concept
Finding the determinant of a 3x3 square matrix or a larger square matrix can involve a lot of computation. To reduce the amount of computation, we can use methods for simplifying determinants. These methods for simplifying determinants involve using row or column operations to change some entries of the matrix to zeros.
Computing determinants can be really complicated when you're dealing with 3 by 3 determinants or higher and so you definitely want to able to simplify a determinant before computing it and there are 3 rules that allow you to do so.
Here is the first, with any determinant you can factor a constant from a row or column so for example here I've got a lot of common factors in each my rows take a look at the last row I have a common factor of 10 you can pull that right out and put it in front so when I compute this determinant instead I can compute this simpler determinant and just multiply the result by 10 and you'll notice I could actually factor more, I can factor 16 out of the top row then I get 160 times and so on and so forth then keep doing that until your determinant becomes nice and simple.
The second thing you could do, you can add a multiple of one row to another and the same goes for columns, so for example you probably notice before that we like to expand along rows or columns with a lot of zeros where you can create more zeros by cleverly adding multiples of one row or column to another. Now here what I've done is I've multiplied the first column by 4 and added it to the third column so this column is now 4 times c1 plus c3 right, 4 times 16 is 64 add that to this negative 64 and you get 0 we do that to the whole column. Now what's interesting about this row operation it's a column operation in this case is that doesn't change the value of the determinant. You can always add a multiple of a row to another row or a multiple of a column to another column and it doesn't change the value.
And the third thing you can do is interchange two rows or two columns, so here for example say for some reasons I want to have this 0 on the left, I can just switch the first two columns so these two columns have been switched. Whenever you switch two rows or two columns, it changes the sign of the determinant, so the 3 things you can do to determinant to simplify; one is you can factor a constant out of any row or any column, two you can add any multiple of one row to another row and the same goes with columns and three you can interchange two columns or two rows, just remember to change the sign when you do that.