Question

Dadas as matrizes A e B , calcule o valor de para que o produto dos determinantes das matrizes

Answer 1

Calculando o determinante de $\mathbf{A},$ sendo

$\mathbf{A}=\left[\begin{array}{ccc} 3&1&2\\0&2&0\\4&1&1 \end{array}\right]\\\\\\ \det\mathbf{A}=3\cdot 2\cdot 1+0+0-4\cdot 2\cdot 2-0-0\\\\ \det\mathbf{A}=6-16\\\\ \boxed{\begin{array}{c}\det\mathbf{A}=-10 \end{array}}$


Calculando o determinante de $\mathbf{B},$ sendo

$\mathbf{B}=\left[\begin{array}{ccc} m&5&1\\3&1&7\\0&0&3 \end{array}\right]\\\\\\ \det\mathbf{B}=m\cdot 1\cdot 3+0+0-0-0-3\cdot 3\cdot 5\\\\ \boxed{\begin{array}{c}\det\mathbf{B}=3m-45 \end{array}}$

_________________

Queremos que o produto dos determinantes seja igual a 330:

$\det\mathbf{A}\cdot \det\mathbf{B}=330\\\\ -10\cdot (3m-45)=330\\\\ 3m-45=\dfrac{330}{-10}\\\\\\ 3m-45=-33\\\\ 3m=-33+45\\\\ 3m=12\\\\ m=\dfrac{12}{3}\\\\\\ \boxed{\begin{array}{c}m=4 \end{array}}$


Bons estudos! :-)