Question

Dadas as matrizes A= [ 2 1 ] [1 1] e M= [1 0] [ 2 1].
a) determine M-¹.
b) sabendo que o traço de uma matriz é a soma dos elementos da diagonal principal, determine o traço da matriz
M-¹ . A . M.

Answer 1

Resposta:

Explicação passo-a-passo:

a) Matriz M⁻¹ = M.X = I, sendo

$X=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$

Assim

$M.X=I=>\left[\begin{array}{cc}1&0\\2&1\end{array}\right].\left[\begin{array}{cc}a&b\\c&d\end{array}\right]=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]=>\left[\begin{array}{cc}a+0&b+0\\2a+c&2b+d\end{array}\right]=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]$

Logo

a = 1

b = 0

2a+c = 0 => 2.1+c = 0 => c = -2

2b+d = 1 => 2.0+d = 1 => d = 1

Assim

$M^{-1}=\left[\begin{array}{cc}1&0\\-2&1\end{array}\right]$

b)

M⁻¹.A.M = M.M⁻¹.A = I.A =>

$I.A = \left[\begin{array}{cc}1&0\\0&1\end{array}\right].\left[\begin{array}{cc}2&1\\1&1\end{array}\right]=\left[\begin{array}{cc}2+0&1+0\\0+1&0+1\end{array}\right]=\left[\begin{array}{cc}2&1\\1&1\end{array}\right]$

Assim, o traço da matriz M⁻¹.A.M é: 2 + 1 = 3