Question on matrices

Question

Let A be a symmetric matrix of form 2.2 then what should be the elements in matrix B of2.2 such that AB not equals to BA

Answer 1

In general , a matrix does not commute with its conjugate: if $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, then $$A {\,}^{\mathrm t\!}A =\begin{bmatrix}a^2+b^2&ac+bd\\ac+bd&b^2+d^2\end{bmatrix},\qquad {\,}^{\mathrm t\!}A\,A =\begin{bmatrix}a^2+c^2&ab+cd\\ab+cd&c^2+d^2\end{bmatrix}.$$ They're different as soon as $A$ is not symmetric.

Answer 2

hint

Let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $B = \begin{pmatrix} x & y \\ z & w \end{pmatrix}$. Notice that $$ AB = \begin{pmatrix} ax+bz & ay+bw \\ cx+dz & cy+dw \end{pmatrix} $$ Can you work out what $BA$ should be and then pick some simple values to make sure $AB \neq BA$?

Answer 3

Write $$A = \begin{bmatrix} a & b\\ c & d\\ \end{bmatrix}$$

and $$B = \begin{bmatrix} e & f\\ g & h\\ \end{bmatrix}$$

Then

$$ AB = \begin{bmatrix} ae+bg & af + bh\\ ce + dg & cf + dh \end{bmatrix}\\ BA = \begin{bmatrix} ae+cf & be + df\\ ag + ch & bg + dh \end{bmatrix}\\ $$

From this we get that either one of these must hold: $bg \neq cf$, $af+bh \neq be+df$ or $ag+ch \neq ce + dg$.