Question

Determine o valor de x para que o determinante da matriz A seja igual a 6.

| x _____-3 |

| x+2__ x-2 |


a) 1

b) -2 e 1

c) 1 e 0

d) -1 e2

e) -2

Answer 1

Desejamos calcular os valores de x para que o determinante da matriz A seja igual 6, sendo a matriz A igual a:

$\Large\displaystyle\begin{gathered}\sf A_{2\times 2}=\left[\begin{array}{ccc}\tt x&\tt -3\\\tt x+2&\tt x-2\end{array}\right] \end{gathered}$

Sabendo disso, temos que:

$\Large\displaystyle\begin{gathered}\sf \det \left(A_{2\times 2}\right)=\left|\begin{array}{ccc}\tt x&\tt -3\\\tt x+2&\tt x-2\end{array}\right|=6 \end{gathered}$

E com isso, temos que:

$\Large\displaystyle\begin{gathered}\sf x\cdot(x-2)-\left[ -3\cdot(x+2)\right]=6 \end{gathered}$

$\Large\displaystyle\begin{gathered}\sf x^2-2x-\left[ -3x-6\right]=6 \end{gathered}$

$\Large\displaystyle\begin{gathered}\sf x^2+x+6=6 \end{gathered}$

$\Large\displaystyle\begin{gathered}\sf x^2+x=0 \end{gathered}$

$\Large\displaystyle\begin{gathered}\sf x\cdot (x+1)=0 \end{gathered}$

$\Large\displaystyle\begin{gathered}\sf \boxed{\tt x=0}\ \ \wedge\ \ \boxed{\tt x=-1 } \end{gathered}$

Portanto os valores de x para que o det da matriz A seja igual a 6 são respectivamente 0 e -1.