Question

Daca AB=BA,aratati că det B mai mic sau egal cu 0. Matricele A= (1 2 4 1)
B=(0 x y 0)

Answer 1

$\displaystyle A= \left(\begin{array}{ccc}1&2\\4&1\\ \end{array}\right) \\ \\ B= \left(\begin{array}{ccc}0&x\\y&0\\ \end{array}\right) \\ \\ AB=BA$
$\displaystyle AB=\left(\begin{array}{ccc}1&2\\4&1\\ \end{array}\right) \cdot \left(\begin{array}{ccc}0&x\\y&0\\ \end{array}\right) = \left(\begin{array}{ccc}1 \cdot 0+2 \cdot y&1 \cdot x+2 \cdot 0\\4 \cdot 0+1 \cdot y&4\cdot x+1 \cdot 0 \\ \end{array}\right)= \\ \\ =\left(\begin{array}{ccc}2y&x\\y&4x\\\end{array}\right)$
$BA=\left(\begin{array}{ccc}0&x\\y&0\\ \end{array}\right) \cdot \left(\begin{array}{ccc}1&2\\4&1\\ \end{array}\right)=\left(\begin{array}{ccc}0 \cdot 1+x \cdot 4 &0 \cdot 2 +x \cdot 1\\y \cdot 1+0 \cdot 4&y \cdot 2+0\cdot1\\\end{array}\right) = \\ \\ =\left(\begin{array}{ccc}4x&x\\y&2y\\\end{array}\right)$
$2y=4x \Rightarrow y=2x \\ \\ det~B= \left|\begin{array}{ccc}0&x\\2x&0\\\end{array}\right|=0 \cdot 0-x \cdot 2x=0-2x^2=-2x^2 \leq 0$