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Algebra A – Learning Log 2

Definitions:

Let \(A\) be a matrix of order \(n \times n\).

  1. The trace of \(A\) is denoted by \(\text{tr}(A)\), and is equal to the sum of the elements on its main diagonal:

\(\text{tr}(A) = a_{11} + a_{22} + \dots + a_{nn}\)

  1. \(A\) is called a scalar matrix if it is diagonal, and all the elements on its main diagonal are equal:

\(A = \lambda \cdot I = \begin{pmatrix} \lambda & 0 & \dots & 0 \\ 0 & \lambda & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & \lambda \end{pmatrix}\)

Answer the following problem:

(a) Let \(A\) be a matrix of order \(2 \times 2\) satisfying \(\text{tr}(A) = 0\). Prove that \(A^2\) is a scalar matrix.

Since \(\text{tr}(A)=0\Rightarrow a_{11}=-a_{22}\), then \(A = \begin{pmatrix} a & b \\ c & -a \end{pmatrix}\)

Then \(A^2 = \begin{pmatrix} a & b \\ c & -a \end{pmatrix} \begin{pmatrix} a & b \\ c & -a \end{pmatrix}\)​\(= \begin{pmatrix} a^2 + bc & 0 \\ 0 & a^2 + bc \end{pmatrix}\)​\(= \begin{pmatrix} a^2 + bc \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

Thus \(A^2\) is a scalar matrix.

(b) Let \(A\) and \(B\) be matrices of order \(2 \times 2\). Prove that \(\text{tr}(AB - BA) = 0\).

Since \(A=\begin{pmatrix}a & b\\ c & d\end{pmatrix},B=\begin{pmatrix}w & x\\ y & z\end{pmatrix}\), then \(AB=\begin{pmatrix}a & b\\ c & d\end{pmatrix}\begin{pmatrix}w & x\\ y & z\end{pmatrix}=\begin{pmatrix}aw+by & ax+bz\\ cw+\mathrm{d}y & cx+dz\end{pmatrix}\)

\(BA=\begin{pmatrix}w & x\\ y & z\end{pmatrix}\begin{pmatrix}a & b\\ c & d\end{pmatrix}=\begin{pmatrix}wa+cx & wb+\mathrm{d}x\\ ay+cz & by+dz\end{pmatrix}\)

Then \(AB-BA=\begin{pmatrix}aw+by & ax+bz\\ cw+\mathrm{d}y & cx+dz\end{pmatrix}-\begin{pmatrix}wa+cx & wb+\mathrm{d}x\\ ay+cz & by+dz\end{pmatrix}=\begin{pmatrix}by-cx & ax-\mathrm{d}x-bw+bz\\ -ay+\mathrm{d}y+cw-cz & cx-by\end{pmatrix}\)

Thus \(\text{tr}(AB-BA)=by-cx+cx-by=0\)

(c) Let \(A\), \(B\), and \(C\) be matrices of order \(2 \times 2\). Prove that \(C(AB - BA)^2 = (AB - BA)^2 C\).

Let \(C=\begin{pmatrix}e & f\\ g & h\end{pmatrix}\) and we have know that \(AB-BA=P=\begin{pmatrix}p & q\\ r & s\end{pmatrix}\)

Since \(tr(AB-BA)=0\), then \(P=\begin{pmatrix}p & q\\ r & -p\end{pmatrix}\Rightarrow P^2=\begin{pmatrix}p^2+qr\end{pmatrix}\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\)

Hence \(CP^2=\begin{pmatrix}e & f\\ g & h\end{pmatrix}\begin{pmatrix}p^2+qr\end{pmatrix}\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}=\left(p^2+qr\right)\begin{pmatrix}e & f\\ g & h\end{pmatrix}\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}=\left(p^2+qr\right)\begin{pmatrix}e & f\\ g & h\end{pmatrix}\)

\(P^2C=\begin{pmatrix}p^2+qr\end{pmatrix}\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}\begin{pmatrix}e & f\\ g & h\end{pmatrix}=\left(p^2+qr\right)\begin{pmatrix}e & f\\ g & h\end{pmatrix}\)

Hence \(CP^2=P^2C\), thus \(C(AB - BA)^2 = (AB - BA)^2 C\)

Question: Can these results be extended to matrices of order \(n \times n\)? Add an explanation to support your answer.

No, \(\left(AB-BA\right)^2\) must be a scalar matrix, which means \(AB-BA\) must satisfy \(a_{11}=-a_{22}\), then \(|a_{11}|=|a_{22}|\)

But if \(n>2\), there are more than 2 elements in the main diagonal and can't satisfy \(|a_{11}|=|a_{22}|\)

\(CP^2=\begin{pmatrix}e & f\\ g & h\end{pmatrix}\begin{pmatrix}p^2+qr\end{pmatrix}\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}=\left(p^2+qr\right)\begin{pmatrix}e & f\\ g & h\end{pmatrix}\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}=\left(p^2+qr\right)\begin{pmatrix}e & f\\ g & h\end{pmatrix}\)

Answer the following questions:

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

Options: 1 2 3 4 5 6 7 8 9

  1. How confident are you that you answered correctly, on a scale of 1-9? (1 - not confident at all, 9 - very confident)

Options: 1 2 3 4 5 6 7 8 9

  1. If you think you were right, what helped you answer correctly? If you think you were wrong, what could have helped you answer correctly?

    The problem a and b hint me how to deal with problem c. They have the similar form but at the first time it is hard to discover. And if i consider that: the problem given me above must have some function, then i finally discover its function.

  2. What skills are necessary in order to succeed in the course?

    Generalization ability. We need it to generalize a more general method or thought to deal with a lot of questions based on fewer samples. And this also promotes our efficiency.

    Ability to observe questions effectively, Sometimes the hint of questions is in themselves. We need cultivate this ability to let us solve the problem.