Question

Determine a inversa da matriz A=(a11=3, a12=4, a21=1, a22=0)

Answer 1

Sua questão:

Determine a inversa da matriz A=(a11=3, a12=4, a21=1, a22=0)

Resolução:

$\large\sf \left(\begin{array}{ccc}3&4\\1&0\end{array}\right) \cdot \left(\begin{array}{ccc}a&b\\c&d\end{array}\right) = \left(\begin{array}{ccc}1&0\\0&1\end{array}\right)$

$\large\sf \left(\begin{array}{ccc}3*a +4*c&3*b+4*d\\1*a+0*c&1*b+0*d\end{array}\right) = \left(\begin{array}{ccc}1&0\\0&1\end{array}\right)$

Perceba que toda multiplicação que envolva o zero sempre resultará em zero então:

$\large\sf \left(\begin{array}{ccc}3*a +4*c&3*b+4*d\\1*a+\not{0*c}&1*b+\not{0*d}\end{array}\right) = \left(\begin{array}{ccc}1&0\\0&1\end{array}\right)$

$\large\begin{array}{lr}\sf 1*a = 0\\\\\sf \underline{\boxed{\red{\sf a= 0}}}\\\\\\\sf 1b =1\\\\\sf \underline{\boxed{\red{\sf b = 1}}}\\\\\\\sf 3*a + 4*c =1\\\\\sf 3*0 + 4*c =1\\\\\sf 4c =1\\\\\sf \underline{\boxed{\red{\sf c = \dfrac{1}{4}}}}\\\\\\\sf 3*b + 4*d = 0\\\\\sf 3*1 + 4*d = 0\\\\\sf 3+4d=0\\\\\sf 4d =-3\\\\\sf \underline{\boxed{\red{\sf d=\frac{-3}{4} }}} \end{array}$

Portanto a matriz inversa da matriz A é:

$\large\left(\begin{array}{ccc}0&1\\1/4&-3/4\end{array}\right)$

Espero ter ajudado.

Bons estudos.

• Att. FireClassis