$$

Midterm 1: Questions to ponder

Please see Course website for details about the exam.

You need to provide the reasons for your answers.

How to prepare the exam?

1. Make a concrete setting for the problems whenever possible. For example, what is the length of
   $(1,2,3)$?
2. Write down your answer and reasons on a paper.
3. If you do not know how to answer it, look it up from the following resources, or ask ChatGPT.
   • Hefferon Linear Algebra by Jim Hefferon
   • NB Linear algebra notebook by Jephian Lin
   • LR Learning resources
4. Think carefully whether your answer is correct or not.
5. Repeat the above steps with different settings.

Calculation of determinant

• How to calculate the determinant of
  $A$ by definition?
• How to calculate the determinant of
  $A$ by Laplace expansion?
• How to calculate the determinant of
  $A$ by permutation expansion?
• How to write an invertible matrix as the product of elementary matrices?
• What is the determinant of an elementary matrix?

Related resources: Hefferon Four.I.1~3, III, LR 3~8, 12, 13, NB 401, 402, 405, 409, 412, 413

Properties of determinant

• Why
  $det(A)$ is the volume of the parallelotope?
• Why
  $det(A)=0$ when
  $A$ contains dependent rows?
• How to determine if
  $A$ is invertible by
  $det(A)$?
• How to derive the distributive law from the definition?
• Why
  $det(A)=det(A^{\mathrm{\top }})$?
• Why
  $det(AB)=det(A)det(B)$?
• Why
  $det\left[\begin{array}{cc}A& B\\ O& D\end{array}\right]=det(A)det(D)$?
• Is
  $det\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]=det(A)det(D)-det(B)det(C)$? Prove it or find a countnerexample.
• How to use the distributive law to derive the Laplace expansion formula and the permutation formula?

Related resources: Hefferon Four.I.3, II, III , LR 9~11, 14, 15, NB 404, 406, 407, 408, 409, 412, 413

This note can be found at Course website > Learning resources.