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 \(\displaystyle \begin{bmatrix}p & q \\r & s \end{bmatrix} \in \mathcal{B}\).