Question

as matrizes A=(1 2 3 0) e B= ( x y 3 5) comutam, isto é, são tais que A.B=B.A. Calcule x e y

Answer 1

$A=\begin{pmatrix}1&2\\3&0\end{pmatrix}$

$B = \begin{pmatrix}x&y\\3&5\end{pmatrix}$

$A.B = \begin{pmatrix}1.x+2.3&1.y+2.5\\3.x+0.3&3.y+0.5\end{pmatrix}\\A.B=\begin{pmatrix}x+6&y+10\\3x&3y\end{pmatrix}$

$B.A=\begin{pmatrix}x.1+y.3&x.2+y.0\\3.1+5.3&3.2+5.0\end{pmatrix}\\B.A=\begin{pmatrix}x+3y&2x\\18&6\end{pmatrix}$

Como as matrizes comutam, podemos afirma que

$\mathsf{A.B=B.A}\\\begin{pmatrix} x+6&y+10\\3x&3y\end{pmatrix}=\begin{pmatrix}x+3y&2x\\18&6\end{pmatrix}$

Pela igualdade de matrizes podemos escrever

$\begin{cases}\mathsf{3x=18}\\\mathsf{3y=6}\end{cases}$

$\mathsf{3x=18}\\\mathsf{x=\dfrac{18}{3}}\\\large\boxed{\boxed{\boxed{\mathsf{x=6}}}}$

$\mathsf{3y=6}\\\mathsf{y=\dfrac{6}{3}}\\\large\boxed{\boxed{\boxed{\mathsf{y=2}}}}$

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