Lecture 19: Introduction to Determinants (Nicholson Section 3.1)
Sub Project: Linear Algebra Lecture Videos
Alternate Video Access via MyMedia | Video Duration: 40:57
Description: Is the general 2x2 matrix A = [a,b;c,d] invertible? Used the matrix inversion algorithm on this general 2x2 matrix and found that in order for to be invertible we need ad-bc to be nonzero. Also, if ad-bc is nonzero then there’s a formula that we can memorize that gives us the inverse of A.
7:55 --- Stated a theorem for 2x2 matrices about whether or not they’re invertible.
8:30 Defined the determinant of a 2x2 matrix.
12:00 --- Defined the determinant of a 3x3 matrix using a formula. I do not have this formula memorized even though I’ve been using and teaching linear algebra for over 30 years. The reason I could write it down so quickly is because I was looking at the matrix and writing it down by knowing the definition in terms of cofactors (see 15:55 for 3x3 matrices and 29:10 for general NxN matrices) and applying that definition in real time.
14:25 --- For a general 3x3 matrix: if the third row is a multiple of the second row, showed that the determinant is zero. (You should make sure that you can also repeat this argument if the second row is a multiple of the third row.)
15:55 --- Noted that the determinant of a 3x3 matrix is found using a specific linear combination of determinants of 2x2 submatrices.
17:45 --- General discussion of computing determinants of 4x4 matrices, 5x5 matrices, 6x6 matrices --- how many 2x2 determinants will be needed? Computing a determinant’s a lot of work! (Will there be a faster way? We’ll see in the next lecture that there is.)
21:10 --- Defined the determinant of an NxN matrix in terms of a cofactor expansion along the first row. Defined what a cofactor is.
25:45 --- Computed the determinant of a specific 3x3 matrix.
29:00 --- How to use wolfram alpha to find the cofactors of a square matrix; you can use this to check your work.
31:55 --- Stated a theorem that states that the determinant of a square matrix can be computed by using a cofactor expansion about any row or column --- it doesn’t have to just be the first row. You can choose whatever row or column that makes it easiest for you.
33:50 --- Demonstrated the usefulness of this theorem by computing the determinant of an upper triangular matrix.
38:05 --- Theorem: If A is upper triangular or lower triangular or diagonal matrix then det(A) is the product of the diagonal entries of A.
39:25 --- Gave a 3x3 motivation for det(A) = det(A^T).
