Workshop: Determinants


On Friday, we will have the first of perhaps several WORKSHOPS in class. Workshops will ask you to acquire basic knowledge of a topic outside of class through the use of screencasts and other video resources along with guided practice to help you acquire basic skills. Then, in class, instead of spending time on lecture, we will work exercises from the Lay textbook and of the professor's own making in groups. Workshops will have point values assigned to them and will count as Take Home Assessments (even though you may end up taking nothing home). In other words, workshops "flip" or invert the classroom setting so that the lecture takes place outside of class -- where you can pause, rewind, stop and start again, etc. and are under no pressure to unders -- and the harder tasks usually relegated to homework are done in class where you can get help immediately.

Here's how the workshop works:
  • Right now, read the Overview, scan the list of screencasts and read the notes, and read through the list of Competencies.
  • Between now and Tuesday evening, watch the screencasts linked below and work the Guided Practice tasks. NOTE that you will need to view the screencasts in short order and complete one of the Guided Practice tasks before Tuesday afternoon.
  • Before Friday's class, finish the rest of the Guided Practice; re-watch the screencasts to make sure you have attained the Competencies listed; and ask questions in office hours.
  • On Friday, we will:
    • Start with a 5-minute multiple choice quiz over the basic material from the screencasts and Guided Practice. This will be done with clickers so that you will know immediately how you did, and any frequently-missed questions will be used for Q&A following the quiz.
    • Follow the quiz up with a brief Q&A session on anything you need clarification on.
    • Work on exercises and problems related to the material from the screencasts. These will be done in groups of two and will carry a point value. You will be asked to turn in a rough draft of your work at the end of class, and then a final draft will be due on Monday. The entire package -- quiz and problem set -- will be counted as a Take Home Assessment.

This is pretty obviously an attempt to map the inverted classroom methodology from CMP 150 onto MAT 233. I think it could be very beneficial for you. I'll be following up with you later on how this all goes, and if you like it, we'll do more of it.

Overview:

The workshop coming on Friday will be on the important subject of determinants. The determinant of a square matrix is a number that is computed using the individual entries of the matrix -- and a somewhat convoluted formula -- that, among other things, helps us to determine whether a square matrix is invertible. Prior to the workshop, you will watch some video and learn how to calculate -- by hand -- the determinant of a 2x2, 3x3, 4x4, and general nxn matrix in a couple of different ways. In the workshop, we will practice these calculations and discover some important properties of determinants for upper- and lower-triangular matrices and some results about how determinants interact with matrix arithmetic.

Video to watch:

Watch these three videos from the Khan Academy website:

This totals up to a little less than 40 minutes of video to watch. Notes on these videos:
  • The first video on 3x3 determinants assumes that you know how to calculate the inverse of a 2x2 matrix. If you have forgotten, you can review from the book, or watch the video Formula for 2x2 Inverse.
  • If you like these videos, Khan Academy has literally hundreds more, including an entire course on linear algebra in 10-20 minute video segments. Other students in the past have found these very helpful in understanding difficult concepts or procedures.

Competencies:

By class time on Friday, each student should be able to perform the following tasks and respond correctly on a 5-minute quiz:
  • Calculate the determinant of a 2x2 matrix.
  • Calculate the determinant of a 3x3 matrix.
  • Given a square matrix A, a row number i, and a column number j, identify the submatrix A_{ij}.
  • Calculate the determinant of a 4x4 matrix.
  • Calculate the determinant of a square matrix using a cofactor expansion along any row and any column.
  • Use the notation for determinants correctly (either det(A) or |A|). Explain the difference between A and |A|.
  • Explain what the value of a determinant of a matrix tells you about that matrix (in terms of its invertibility).


Exercises


Task 1: Calculate the determinant of the following matrix, by hand, and showing all work. (You can check your work in MATLAB, provided you can figure out how to take determinants in MATLAB.)




Task 2: Consider the matrix A from Task 1. Without doing any further calculations whatsoever, not even arithmetic, answer the following questions:
  • Are the columns of A linearly independent?
  • How many solutions does the system Ax = 0 have?
  • Do the columns of A form a basis for R3?

Give a one-sentence explanation that explains unambiguously why your answers to these questions are correct. (Hint: What previous, lengthy theorem have we seen that ties these three questions together? How can you tell just from the results of Task 1 that the matrix A satisfies the conditions of that theorem?)

Task 3: Calculate the determinant of the following matrix, by hand, and showing all work. Again, you may check your work in MATLAB when done.




Specifications

Here is what to turn in, how, and when:
  • By Tuesday, April 5, at 4:30 PM, hand in your work on Task 1. This can be handwritten as long as it's neat. Note that you will need to have at least watch the first video and done some practice along with it in order to finish this task.
  • By Friday, April 8, at class time, hand in written work for Tasks 2 and 3. Again, handwritten is OK as long as it's neat.

NO LATE WORK WILL BE ACCEPTED.

Lastly: You may ask me (Prof. Talbert) questions on this material AT ANY TIME and THROUGH ANY MEDIUM you wish. I am a resource to help you navigate through the video material and help you translate this into understanding. Please use me!