conditions on Matrix elements

[logistic_guy]

Let $\displaystyle \mathcal{A}$ be the set of $\displaystyle 2 \times 2$ matrices with real number entries. Recall that matrix multiplication is defined by

$\displaystyle \begin{bmatrix}a & b \\c & d \end{bmatrix} \begin{bmatrix}p & q \\r & s \end{bmatrix} = \begin{bmatrix}ap + br & aq + bs \\cp + dr & cq + ds \end{bmatrix}$

Let $\displaystyle M = \begin{bmatrix}1 & 1 \\0 & 1 \end{bmatrix}$

and let $\displaystyle \mathcal{B} = \{X \in \mathcal{A} \ | \ MX = XM\}$

Find conditions on $\displaystyle p, q, r, s$ which determine precisely when $\displaystyle \begin{bmatrix}p & q \\r & s \end{bmatrix} \in \mathcal{B}$.

 

[khansaheb]

Let $\displaystyle \mathcal{A}$ be the set of $\displaystyle 2 \times 2$ matrices with real number entries. Recall that matrix multiplication is defined by

$\displaystyle \begin{bmatrix}a & b \\c & d \end{bmatrix} \begin{bmatrix}p & q \\r & s \end{bmatrix} = \begin{bmatrix}ap + br & aq + bs \\cp + dr & cq + ds \end{bmatrix}$

Let $\displaystyle M = \begin{bmatrix}1 & 1 \\0 & 1 \end{bmatrix}$

and let $\displaystyle \mathcal{B} = \{X \in \mathcal{A} \ | \ MX = XM\}$

Find conditions on $\displaystyle p, q, r, s$ which determine precisely when $\displaystyle \begin{bmatrix}p & q \\r & s \end{bmatrix} \in \mathcal{B}$.

show us your effort/s to solve this problem.

 

[logistic_guy]

$\displaystyle M\begin{bmatrix}p & q \\r & s \end{bmatrix} = \begin{bmatrix}1 & 1 \\0 & 1 \end{bmatrix} \begin{bmatrix}p & q \\r & s \end{bmatrix} = \begin{bmatrix}p + r & q + s \\r & s \end{bmatrix}$


$\displaystyle \begin{bmatrix}p & q \\r & s \end{bmatrix}M = \begin{bmatrix}p & q \\r & s \end{bmatrix} \begin{bmatrix}1 & 1 \\0 & 1 \end{bmatrix} = \begin{bmatrix}p & p + q \\r & r + s \end{bmatrix}$

We want:

$\displaystyle M\begin{bmatrix}p & q \\r & s \end{bmatrix} = \begin{bmatrix}p & q \\r & s \end{bmatrix}M$

Then,

$\displaystyle \begin{bmatrix}p + r & q + s \\r & s \end{bmatrix} = \begin{bmatrix}p & p + q \\r & r + s \end{bmatrix}$

$\displaystyle p + r = p$, means $\displaystyle r = 0$
$\displaystyle q + s = p + q$, means $\displaystyle s = p$

These are the conditions to let $\displaystyle \begin{bmatrix}p & q \\r & s \end{bmatrix} \in \mathcal{B}$.

 

[barbsabbs]

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