Question

A pergunta é:

Verifique que a matriz
$\left[\begin{array}{ccc}cos \alpha &sen \alpha &0\\-sen \alpha &cos \alpha &0\\0&0&1\end{array}\right]$

é invertivel para todo Alpha?

Mim ajudem

Answer 1

$det.matrix = \left[\begin{array}{ccc}cos \alpha &sen \alpha &0\\-sen \alpha &cos \alpha &0\\0&0&1\end{array}\right] \left[\begin{array}{ccc}cos \alpha &sen \alpha \\-sen \alpha &cos \alpha \\0&0\end{array}\right]$

$det.matrix = cos ^{2} \alpha +0+0-(-sen ^{2} \alpha +0+0)$
$det.matrix = cos ^{2} \alpha+sen ^{2} \alpha$
$det.matrix = 1$

$Adj.A = \frac{1}{det.matrix}. adj. A$

adj A :
C1.1 = cos α       C1.2 = sen α    C1.3 = - sen α
C2.1 = - Sen α    C 2.2 = cos α    C2.3 = 0
C3.1 = 0             C 3.2 = 0          C3.3  = 1

Adj A = $A^{-1}= \left[\begin{array}{ccc}cos \alpha &-sen \alpha &0\\sen \alpha &cos \alpha &0\\-sen \alpha &0&1\end{array}\right]$