Section 3.5 Assignment
Activity 3.5.1.
In this activity, we fix a matrix \(A\) first.
Let \(A = \begin{pmatrix}
1 \amp 2\\
0 \amp 1
\end{pmatrix}.\) Find all matrices \(X = \begin{pmatrix}
a \amp b\\
c \amp d
\end{pmatrix}\) such that
\begin{equation*}
AX=XA\text{.}
\end{equation*}
Here is a tough question for you to crack. Without a software, it would take forever to solve this question! In contrast, with a software, you focus on the understanding, not the computation.
Activity 3.5.2.
We assume that none of \(x,y,z\) and \(w\) equals 0. Let \(A = \begin{pmatrix}
x \amp y\\
z \amp w
\end{pmatrix},\) Find all matrices \(X = \begin{pmatrix}
a \amp b\\
c \amp d
\end{pmatrix}\) such that
\begin{equation*}
AX=XA\text{.}
\end{equation*}
Note that we have a linear system, whose variables are \(a,b,c\) and \(d\text{.}\)
\begin{equation*}
\begin{aligned}
0a -zb + yc + 0d \amp= 0 \\
-ya+(x-w)b + 0c + yd \amp= 0 \\
za + 0b + (w-x)c - zd \amp= 0\\
0a +zb -yc + 0d \amp =0
\end{aligned}
\end{equation*}
Question: write down the coefficient matrix \(C\) of the linear system above.
Question: Compute the reduced echelon form of the matrix\(C\text{.}\)
Question: Is it necessary to assume that none of \(x,y,z\) and \(w\) equals 0?
Activity 3.5.3.
Let \(A = \begin{pmatrix}
1 \amp 2\\
3 \amp 4
\end{pmatrix}.\) Find all matrices \(X = \begin{pmatrix}
a \amp b\\
c \amp d
\end{pmatrix}\) such that
\begin{equation*}
AX=XA\text{.}
\end{equation*}
