Minors
and Cofactors: Continuing... (b) To expand along the third column, I need to find the minors and then the cofactors of the thirdcolumn entries: a_{1,3}, a_{2,3}, a_{3,3}, and a_{4,3}.
Wait a sec... The a_{2,3}entry of the original matrix is zero. This means that I'll be getting zero for that term when I expand down the column, no matter what the value of the minor M_{2,3} turns out to be. So I don't really care what the A_{2,3} cofactor is; I can just put "0" for this entry, because a_{2,3}A_{2,3} = (0)(A_{2,3}) = 0. In fact, I can ignore each of the last three terms in the expansion down the third column, because the third column's entries (other than the first entry) are all zero. So the only computation I care about is one I've already done (for part (a), when expanding across the first row): det(A)
= a_{1,3}A_{1,3}
(c) Comparison: The value of the determinant is the same in each expansion. In the example above, we expanded by taking the 4by4 matrix down to 3by3 determinants. But technically, you're "supposed" to go down to 2by2 determinants when you "expand" by this method. That is, the above cofactor "should" have been computed using many more steps. Suppose you'd gone across the first row again. The first cofactor "should" have been computed like this:
By crossing out a row and column, a new matrix is formed. Call this new matrix "C". To find the determinant of C, expand along the first row.
Then the value of det(C) is given by: (1)(0) + (0)(–1) + (–1)(0) = 0 + 0 + 0 = 0 And then: Copyright © Elizabeth Stapel 20062011 All Rights Reserved A_{1,1} = (–1)^{2}det(C) = (1)(0) = 0 And that's just one cofactor; you still have three to go! Using this methodology, all determinants can be boiled down to finding 2by2 determinants. But since you know how to find 3by3 determinants, there is no need to go all the way down to 2by2's (unless the instructions specifically require that you do). As you can see from the previous example, having a "zerorich" row or column in your determinant can make your life a lot easier. Since you'll get the same value, no matter which row or column you use for your expansion, you can pick a zerorich target and cut down on the number of computations you need to do. Of course, not all matrices have a zerorich row or column. But there is a rule that can help: If you add a multiple of one row (or column) to another row (or column), the value of the determinant will not change. In other words, you can do row operations on determinants, creating a row (or column) with lots of zeroes, and you'll still get the right answer. (You can also just multiply rows  without the adding  or switch rows, but those operations will change the determinant's value. The changes are annoying to keep track of, so try only to do the rowaddition operation.) Here's an example... << Previous Top  1  2  3  Return to Index Next >>



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