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# Midterm 1: Questions to ponder

Please see [Course website](https://www.math.nsysu.edu.tw/~chlin/2024SMath104A/2024SMath104A.html) for details about the exam.

:::success
:bulb: 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_](http://joshua.smcvt.edu/linearalgebra/book.pdf) by Jim Hefferon
- `NB` [_Linear algebra notebook_](https://jephianlin.github.io/LA-notebook/index-en.html) by Jephian Lin
- `LR` [Learning resources](https://hackmd.io/@jephianlin/2024SMath104A-resources)
5. **Think carefully** whether your answer is correct or not.
6. 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 $\det(A) = \det(A\trans)$?
- Why $\det(AB) = \det(A)\det(B)$?
- Why $\det\begin{bmatrix} A & B \\ O & D \end{bmatrix} = \det(A)\det(D)$?
- Is $\det\begin{bmatrix} A & B \\ C & D \end{bmatrix} = \det(A)\det(D) - \det(B)\det(C)$? 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.*