Question

sejam as matrizes:
A =
[-4 3 ]
[ 1 2 ]

B =
[1 0 ]
[-1 3]

Calcule a determinante das seguintes mateizes:

A) A
B) B
C) A + B
D) A - B
E) A + 2B
F) A × B
G) A + l2
H) A transposta​

Answer 1

A determinante de uma matriz 2x2 é calculada por:

$det\left(\begin{array}{ccc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right) = a_{11}\times a_{22} - a_{21}\times a_{12}$

Assim, para as matrizes A e B tais que:

$A = \left[\begin{array}{ccc}-4&3\\1&2\end{array}\right]$

$B = \left[\begin{array}{ccc}1&0\\-1&3\end{array}\right]$

a) det(A):

$det\left(\begin{array}{ccc}-4&3\\1&2\end{array}\right) = -4\times2 - 1\times 3 = -8-3 = -11$

b) det(B):

$det\left(\begin{array}{ccc}1&0\\-1&3\end{array}\right) = 1\times3 - (-1)\times 0 = 3$

c) det(A+B):

$A+B = \left[\begin{array}{ccc}-4&3\\1&2\end{array}\right]+\left[\begin{array}{ccc}1&0\\-1&3\end{array}\right] = \left[\begin{array}{ccc}-3&3\\0&5\end{array}\right]$

$det\left(\begin{array}{ccc}-3&3\\0&5\end{array}\right) = -3\times5-0\times3 = -15$

d) A-B:

$A-B = \left[\begin{array}{ccc}-4&3\\1&2\end{array}\right]-\left[\begin{array}{ccc}1&0\\-1&3\end{array}\right] = \left[\begin{array}{ccc}-5&3\\2&-1\end{array}\right]$

$det\left(\begin{array}{ccc}-5&3\\2&-1\end{array}\right) = -5\times(-1)-2\times3 = 5-6 = -1$

e) A + 2B:

$A+2B = \left[\begin{array}{ccc}-4&3\\1&2\end{array}\right]+\left[\begin{array}{ccc}2&0\\-2&6\end{array}\right] = \left[\begin{array}{ccc}-2&3\\-1&-4\end{array}\right]$

$det\left(\begin{array}{ccc}-2&3\\-1&8\end{array}\right) = -2\times8-(-1)\times3 = -16+3 = -13$

f) AxB:

$A\times B = \left[\begin{array}{ccc}-4&3\\1&2\end{array}\right]\times\left[\begin{array}{ccc}2&0\\-2&6\end{array}\right] = \left[\begin{array}{ccc}(-4*1)+(3*-1)&(-4*0+3*3)\\(1*1+2*-1)&(1*0+2*3)\end{array}\right] =\left[\begin{array}{ccc}-7&9\\-1&6\end{array}\right]$

$det\left(\begin{array}{ccc}-7&9\\-1&6\end{array}\right) = -7\times6 - (-1)\times9 = -42+9 = -33$

g) A+I2:

$A+I_2 = \left[\begin{array}{ccc}-4&3\\1&2\end{array}\right]+\left[\begin{array}{ccc}1&0\\0&1\end{array}\right] = \left[\begin{array}{ccc}-3&3\\1&3\end{array}\right]$

$det\left(\begin{array}{ccc}-3&3\\1&3\end{array}\right) = -3\times3-1\times3 = -9-3 = -12$

h) $A^T$

$A^T = \left[\begin{array}{ccc}-4&3\\1&2\end{array}\right]$

$det\left(\begin{array}{ccc}-4&3\\1&2\end{array}\right) = -4\times2 -3\times1 = -8-3 = -11$

Algumas relações úteis que também ajudariam:

$det(A\times B) = det(A) \times det(B)$

$det(A^T) = det(A)$

$det(k\times A) = k^n \times det(A)$

onde A é uma matriz n x n.