Question

Determine se existir a inversa da matriz : [1 2]
[1 0]

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Answer 1

Boa noite Duda

seja a matriz A

l  1     2  l
l  1     0  l

det(A) = 1*0 - 1*2 = -2

seja a matriz adj(A)

l 0   -2 l
l -1   1 l 

a matriz inversa é 

adj(A)/det(A)

l   0     1 l
l 1/2 -1/2 l

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Answer 2

Temos o seguinte sobre matriz inversa:

$\left[\begin{array}{ccc}1&2\\1&0\\\end{array}\right] \cdot \left[\begin{array}{ccc}a&b\\c&d\\\end{array}\right] = \left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right] \\ \\ \\ \left \{ {{1a+2c=1} \atop {1a +0c=0}} \right. \to \left \{ {{a+2c=1} \atop {a =0}} \right. \\ \\ \left \{ {{1b + 2d=0} \atop {1b + 0d=1}} \right. \to \left \{ {{b + 2d=0} \atop {b =1}} \right.$

Logo temos:

$a + 2c = 1 \to 0 + 2c = 1 \to c = \frac{1}{2} \\ \\ b + 2d = 0 \to 1 + 2d = 0 \to d = - \frac{1}{2}$

Logo a matriz inversa:

$\left[\begin{array}{ccc}a&b\\c&d\\\end{array}\right] = \left[\begin{array}{ccc}0&1\\ \frac{1}{2} &- \frac{1}{2} \\\end{array}\right]$