Question

(FEI-SP) Dadas as matrizes A=( a 2/1 0) e B=(0 1/b-1/2), sabendo que B=A^-1, tem-se que:

a)a+b=1
b)a+b=2
c)a+b=1/2
d)a+b=3/2
e)a+b=5/2

Answer 1

$A = \left[\begin{array}{ccc}a&2\\1&0\end{array}\right]$
$B = \left[\begin{array}{ccc}0&1\\b&-1/2\end{array}\right]$

B = $B = A^{-1}$ ⇒ $A.B = \left[\begin{array}{ccc}1&0\\0&1\end{array}\right]$

$A.B= \left[\begin{array}{ccc}a&2\\1&0\end{array}\right] . \left[\begin{array}{ccc}0&1\\b&-1/2\end{array}\right]$

$A.B= \left[\begin{array}{ccc}0+2b&a-1\\0+0&1+0\end{array}\right] = \left[\begin{array}{ccc}2b&a-1\\0&1\end{array}\right] = \left[\begin{array}{ccc}1&0\\0&1\end{array}\right]$

Daí, temos:
2b = 1 ⇒ b = 1/2
a - 1 = 0 ⇒ a = 1

Então:  a + b = 1 + 1/2 = 3/2 (d)