proof related to matrices - 2

[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 \(\displaystyle \mathcal{B} = \{X \in \mathcal{A} \ | \ MX = XM\}\)

Prove that if \(\displaystyle P, Q \in \mathcal{B}\), then \(\displaystyle P \cdot Q \in \mathcal{B}\) (where \(\displaystyle \cdot\) denotes the usual product of two matrices).

 

[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 \(\displaystyle \mathcal{B} = \{X \in \mathcal{A} \ | \ MX = XM\}\)

Prove that if \(\displaystyle P, Q \in \mathcal{B}\), then \(\displaystyle P \cdot Q \in \mathcal{B}\) (where \(\displaystyle \cdot\) denotes the usual product of two matrices).

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