Lecture 20: Elementary row operations and determinants (Nicholson Section 3.1)

1:40 --- Computed the determinant of a specific 3x3 matrix by doing a cofactor expansion about its second column.

6:00 --- Did elementary row operations on A and carried it to Row Echelon Form. Computed the determinant of the REF matrix. When computing determinants you don’t need to carry the matrix to REF, just to upper triangular form! Then you can use the fact that the determinant of an upper triangular matrix is the product of the diagonal entries.

9:30 --- Stated the effect of each elementary row operation on the determinant of a matrix and explained how to remember these rules.

14:05 --- revisited the previous example and figured out how to figure out the det(A) from the determinant of the REF matrix as long as you know the sequence of elementary row operations you took to get to the REF. If all you have is A and the REF matrix then you can’t find det(A) from the determinant of the REF matrix.

18:10 --- Is there any reason to carry A all the way to RREF? Did this for an example and showed that it still works but it’s an unnecessary amount of work if all you want is det(A). The real point of this example was to show why if det(A) is nonzero then the RREF of A must be the identity matrix (and therefore A is invertible).  And if det(A) is zero then the RREF of A must have a row of zeros (and therefore A is not invertible).

31:25 --- Proved that if you create B by multiplying a row of A by t then det(B) = t det(A). Did this for a 3x3 matrix.

38:00 --- Proved that if create B by swapping two rows of A then det(B) = - det(A). I proved this by induction. I proved that it’s true for 2x2 matrices by using the definition of determinant. I showed how to leverage this knowledge about 2x2 matrices to prove that it’s true for 3x3 matrices. The next step is to leverage this knowledge about 3x3 matrices to prove that it’s true for 4x4 matrices. This goes on forever and this idea is the idea behind proof by induction.

48:40 --- Used the theorem to show that if A has a repeated row then det(A) = 0.