Question

Sabe-se que A=$\left[\begin{array}{ccc}x&1&2\\3&y&5\\2&3&z\end{array}\right]$ ; B=(bij)₃x₃ é uma matriz diagonal  e A×B=$\left[\begin{array}{ccc}2&3&10\\6&12&25\\4&9&20\end{array}\right]$. Qual os valores de x,y,z?

Answer 1

 Olá Ivan,
deves saber o que é uma matriz diagonal: é aquela na qual todos os elementos são nulos, com excessão dos elementos da diagonal principal.

 Daí,

$A\times\,B=\begin{pmatrix}2&3&10\\6&12&25\\4&9&20\end{pmatrix}\\\\\begin{pmatrix}x&1&2\\3&y&5\\2&3&z\end{pmatrix}\times\begin{pmatrix}b_{11}&0&0\\0&b_{22}&0\\0&0&b_{33}\end{pmatrix}=\begin{pmatrix}2&3&10\\6&12&25\\4&9&20\end{pmatrix}$

$\begin{pmatrix}x\cdot\,b_{11}&1\cdot\,b_{22}&2\cdot\,b_{33}\\3\cdot\,b_{11}&y\cdot\,b_{22}&5\cdot\,b_{33}\\2\cdot\,b_{11}&3\cdot\,b_{22}&z\cdot\,b_{33}\end{pmatrix}=\begin{pmatrix}2&3&10\\6&12&25\\4&9&20\end{pmatrix}\\\\\begin{cases}1\cdot\,b_{22}=3\Rightarrow\boxed{b_{22}=3}\\2\cdot\,b_{33}=10\Rightarrow\boxed{b_{33}=5}\\3\cdot\,b_{11}=6\Rightarrow\boxed{b_{11}=2}\end{cases}$

 Por fim,

$\begin{cases}y\cdot\,b_{22}=12\Rightarrow\,y\cdot3=12\Rightarrow\boxed{\boxed{y=4}}\\z\cdot\,b_{33}=20\Rightarrow\,z\cdot5=20\Rightarrow\boxed{\boxed{z=4}}\\x\cdot\,b_{11}=2\Rightarrow\,x\cdot2=2\Rightarrow\boxed{\boxed{x=1}}\end{cases}$