Question

dadas as matrizes A=[-1 3 2 -8] e B=[2 -1 3 0]
a)det (AB)
b)det A. det B
c) det A't

Answer 1

$A\: . \: B \\ \\ \left[\begin{array}{ccc}- 1&3\\2&- 8\end{array}\right] . \left[\begin{array}{ccc}2&- 1\\3&0\end{array}\right] = \left[\begin{array}{ccc}7&1\\- 20&- 2\end{array}\right] \\ \\ \\ det\: A.B = \left|\begin{array}{ccc}7&1\\- 20&- 2\end{array}\right| = 6$

$det\: (A.B) = det\: A\: . \: det\: B \\ \\ det\: (A.B) = \left|\begin{array}{ccc}- 1&3\\2&- 8\end{array}\right| . \left[\begin{array}{ccc}2&- 1\\3&0\end{array}\right] \\ \\ det\: (A.B) = (8 - 6).(0 + 3) = 6$

$A^t = \left[\begin{array}{ccc}- 1&2\\3&- 8\end{array}\right] \\ \\ \\ det\: A^t = \left|\begin{array}{ccc}- 1&2\\3&- 8\end{array}\right| = 8 - 6 = 2$