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Prove or disprove the following claims:

  1. \(A\) and \(B\) are \(5 \times 5\) matrices such that \(AB + BA = 0\). Then at least one of the matrices is not invertible.

    Proof

    Since \(AB + BA = 0\), then \(AB=-BA\).

    Then \(\det AB=\det(-BA)=-\det\left(BA\right)\) since \(A\) and \(B\) are \(5\times 5\) matrices

    Then \(\det A\cdot \det B=-\det A\cdot \det B\), then \(\det A\cdot \det B=0\)​

    Then \(\det A=0\) or \(\det B=0\) or \(\det A=\det B=0\)​

    Thus at least one of the matrices is not invertible.

  2. \(A\) is a square matrix satisfying \(\text{det}(A^3) = \text{det}(A)\). Then \(\text{det}(A^2) = 1\).

    Disproof

    Since \(\det A\cdot \det A^2=\det A\), then \(\det A=0\) or \(\det A^2=1\) when \(\det A\neq 0\)​

    Which is false

Rate the difficulty of the given problem. (1 - very easy, 9 - very hard)

Scale: 1 2 3 4 5 6 7 8 9

What helped you solve the problem?

The use of determinant to cope with the high dimension problem

Suppose the problem is given in the exam, and is worth 20 points. How many points do you think you gained for your solution? Please explain.

\(17\)

Easy problem just use the property of determinant. But maybe need more explanation or quote the content of theorem?

Do you agree with the following statements? (1 - don’t agree at all, 9 - strongly agree)

(a) The purpose of the problem is to evaluate algorithmic skills, such as arithmetic calculations, technical skills, solving methods, etc.

Scale: 1 2 3 4 5 6 7 8 9

(b) The purpose of the problem is to evaluate cognitive skills, such as intuitive understanding, train of thought, strategy, etc.

Scale: 1 2 3 4 5 6 7 8 9

(c) The purpose of the problem is to evaluate formal skills, such as mathematical logic, proof structure, proper writing, etc.

Scale: 1 2 3 4 5 6 7 8 9

Do you agree with the following statements? (1 - don’t agree at all, 9 - strongly agree)

(a) The main purpose of first semester courses in mathematics is to develop algorithmic skills, such as arithmetic calculations, technical skills, solving methods, etc.

Scale: 1 2 3 4 5 6 7 8 9

(b) The main purpose of first semester courses in mathematics is to develop cognitive skills, such as intuitive understanding, train of thought, strategy, etc.

Scale: 1 2 3 4 5 6 7 8 9

(c) The main purpose of first semester courses in mathematics is to develop formal skills, such as mathematical logic, proof structure, proper writing, etc.

Scale: 1 2 3 4 5 6 7 8 9

(a) Do you agree with the following statement? (1 - don’t agree at all, 9 - strongly agree)

The main purpose of first semester courses in mathematics is to learn how to be a student in GTIIT.

Scale: 1 2 3 4 5 6 7 8 9

Explain in your own words, what does “being a student in GTIIT” mean? Is it different than being a student in high school in China? Is it different than being a student in a different university in China?

Yes, usually in other university, even though students study in top university of Shanghai as far as i knew, they don't receive so many hard courses and high demand in study. The teaching and learning here is totally different with others from the length of class to the depth of knowledge, which is a challenge to the fresh man. There is no supervision and we are free to do what we want. We also need to learn how to learn and how to arrange the study.

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