Question

Tendo em vista as matrizes quadradas de ordem 2, sendo M = A•Bt em que Bt é a matriz transposta de B, é correto afirmar que o determinante da matriz inversa de M é

Answer 1

Olá


Resposta correta, Letra C) 1/4


$\displaystyle\mathsf{A= \left[\begin{array}{ccc}1&0\\2&1\\\end{array}\right] }\qquad\qquad\mathsf{B= \left[\begin{array}{ccc}2&1\\0&2\\\end{array}\right] }\\\\\\\\\mathsf{B^t= \left[\begin{array}{ccc}2&0\\1&2\\\end{array}\right] }$


$\displaystyle\mathsf{A\cdot B^t}\\\\\\\\\mathsf{ \left[\begin{array}{ccc}1&0\\2&1\\\end{array}\right] }\cdot \mathsf{ \left[\begin{array}{ccc}2&0\\1&2\\\end{array}\right] }\\\\\\\\\mathsf{ \left[\begin{array}{ccc}(1\cdot 2) +(0\cdot1)\quad &(1\cdot0)+(0\cdot2)\\ (2\cdot2) +(1\cdot 1)\quad&(2\cdot0)+(1\cdot2)\\\end{array}\right] }\\\\\\\\\mathsf{ A\cdot B^t~=~\left[\begin{array}{ccc}2&0\\5&2\\\end{array}\right] }$



Calculando a inversa da matriz M,
Lembrando que a matriz M = A.B^t



Para calcular a matriz inversa 2x2, basta:

1º - Calcular o determinante da matriz

2º - Trocar a posição dos elementos da diagonal principal

3º - Trocar o sinal dos elementos da diagonal secundária

4º - Dividir os elementos da matriz pelo determinante.



$\displaystyle \mathsf{M=\left[\begin{array}{ccc}2&0\\5&2\\\end{array}\right] }}\\\\\\\mathsf{determinante = 4}$

$\displaystyle \mathsf{M=\left[\begin{array}{ccc}2&0\\-5&2\\\end{array}\right] }}\\\\\\\\\text{Divide a matriz pelo determinante.}\\\\\\\\\\\mathsf{M=\left[\begin{array}{ccc} \frac{2}{4} &0\\\\- \frac{5}{2} & \frac{2}{4} \\\end{array}\right] }}}\\\\\\\\ \boxed{\mathsf{M^{-1}=\left[\begin{array}{ccc} \frac{1}{2} &0\\\\- \frac{5}{2} & \frac{1}{2} \\\end{array}\right] }}}$


Por fim, calculando o determinante da matriz inversa de M


$\displaystyle \left|\begin{array}{ccc} \frac{1}{2} &0\\\\- \frac{5}{2} & \frac{1}{2} \\\end{array}\right| }}\\\\\\\mathsf{det=\left( \frac{1}{2}\cdot \frac{1}{2}\right)-\left(-\frac{5}{2}\cdot0\right)}\\\\\\\\\mathsf{det= \frac{1}{4}- 0 }\\\\\\\\\boxed{\mathsf{det=\frac{1}{4}}}\qquad\qquad\Longrightarrow\qquad\text{Letra C}$