Finding the Set of Permutatable Matrices with Algebra

[esmeco]

I have this question as homework from my Algebra class:
A square matrix X is called exchangeable with A if AX=XA.Determine the set of permutable matrices with

.

My question is,how do I find that set?I know that a matrix to be permutable all rows and columns must be the same and that a square matrix is composed by the same number of rows and columns.
Thanks in advance for the help!

 

[HallsofIvy]

?? You defined "exchangeable" with A and then asked for "permutable" with A?? Then you defined "permutable" matrix without any reference to a matrix A?? What am I missing?


If you want to find all matrices that are "exchangeable" with A (standard terminology: "that commute with A"), then look at
$$\left[\begin{array}{cc}a & b \\ c & d \end{array}\right]\left[\begin{array}{cc}1 & 1 \\ 0 & 1\end{array}\right]= \left[\begin{array}{cc}1 & 1 \\0 & 1\end{array}\right]\left[\begin{array}{cc}a & b \\c & d\end{array}\right]$$

If I understand your definition of "permutable" correctly: "all rows and columns must be the same", then all 2 by 2 permutable matrices are of the form
$$\left[\begin{array}{cc}a & a \\ a & a\end{array}\right]$$
and the only "permutable" matrix that is "exchangeable" with A is
$$\left[\begin{array}{cc}0 & 0 \\ 0 & 0 \end {array} \right]$$

 

[esmeco]

...

Sorry for the mistypeing! When I said "exchengeable I meant to say permutable,so it would be like:

"A square matrix X is called permutable with A if AX=XA..."