Question

Me ajudem por favor é sobre matrizes

1. dada a matriz a= (19 19 19
0 0 0
1 1 1
CALCULE (A^2)^T

2. Dada a matriz b= ( 19 19 19
-19 -19 -19
CALCULE (b^t)b

3. Dada a matriz c= ( 19/3 -19/3
-19/3 19/4
CALCULE c-c^t

Answer 1

1) Temos que $A= \left[\begin{array}{ccc}19&19&19\\0&0&0\\1&1&1\end{array}\right]$.

Primeiro, vamos calcular A², ou seja, multiplicar a matriz A por ela mesma:

$A^2= \left[\begin{array}{ccc}19&19&19\\0&0&0\\1&1&1\end{array}\right] . \left[\begin{array}{ccc}19&19&19\\0&0&0\\1&1&1\end{array}\right]$
$A^2= \left[\begin{array}{ccc}380&380&380\\0&0&0\\20&20&20\end{array}\right]$

Para definir a matriz transposta basta trocar as linhas pelas colunas e vice e versa:

$(A^2)^T = \left[\begin{array}{ccc}380&0&20\\380&0&20\\380&0&20\end{array}\right]$

2) Temos que $B= \left[\begin{array}{ccc}19&19&19\\-19&-19&-19\end{array}\right]$

Calculando a matriz transposta:

$B^T= \left[\begin{array}{ccc}19&-19\\19&-19\\19&-19\end{array}\right]$

Portanto,

$B^T.B = \left[\begin{array}{ccc}19&-19\\19&-19\\19&-19\end{array}\right].\left[\begin{array}{ccc}19&19&19\\-19&-19&-19\end{array}\right]$
$B^T.B= \left[\begin{array}{ccc}722&722&722\\722&722&722\\722&722&722\end{array}\right]$

3) Temos que $C=\left[\begin{array}{ccc} \frac{19}{3} &- \frac{19}{3} \\- \frac{19}{3} & \frac{19}{3} \end{array}\right]$

Calculando a matriz transposta:

$C^T=\left[\begin{array}{ccc} \frac{19}{3} &- \frac{19}{3} \\- \frac{19}{3} & \frac{19}{3} \end{array}\right]$

Perceba que as duas matrizes são iguais.

Portanto, $C-C^T = 0$