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Matrix $\textbf{A} = \begin{bmatrix}2&-1\\-1&1\end{bmatrix}.$

Give two examples of a non-zero matrix $\textbf{A}$ such that $\textbf{AX} = \textbf{XA}$?

Find the general form of $\textbf{AX}$?

Michael Hardy
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asdasd
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  • There is the trivial case of the identity matrix. And and then there are matrices which you can find if you know some things about eigenvectors. – Doug M Nov 14 '17 at 20:23
  • Did you for ALL matrices $\mathbf{X} \text{ ?}$ If so, it would be better to say so. $\qquad$ – Michael Hardy Nov 14 '17 at 20:27
  • This might be helpful: https://math.stackexchange.com/questions/170241/when-is-matrix-multiplication-commutative – aleden Nov 14 '17 at 20:33

1 Answers1

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The characteristic and minimal polynomials coincide, $\lambda^2 - 3 \lambda + 1.$ Therefore all matrices that commute with $A$ are polynomials in $A,$ by Cayley-Hamilton we can take these as $$ xI + y A $$

see http://math.stackexchange.com/questions/92480/given-a-matrix-is-there-always-another-matrix-which-commutes-with-it/92832#92832

Or you can write $$ X = \left( \begin{array}{cc} p & q \\ r & s \end{array} \right) $$ and write out the matrix $AX-XA$ and set all four matrix entries to zero. You get simple equations restricting the entries of $X.$

Will Jagy
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