Question

Se A é uma matriz quadrada de ordem 2 definida por a ij =2i-3j sobre i, calcule:
a) a matriz A
b) a matriz At
c) det A

Answer 1

$A=\left[\begin{array}{cc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right]$

a)

$\boxed{a_{ij}=2i-3j}\\\\ a_{11}=2(1)-3(1)\ \therefore\ \boxed{a_{11}=-1}\\\\ a_{12}=2(1)-3(2)\ \therefore\ \boxed{a_{12}=-4}\\\\ a_{21}=2(2)-3(1)\ \therefore\ \boxed{a_{21}=1}\\\\ a_{22}=2(2)-3(2)\ \therefore\ \boxed{a_{22}=-2}$

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

b)

$A^t=\left[\begin{array}{cc}a_{11}&a_{21}\\a_{12}&a_{22}\end{array}\right]\ \therefore\ \boxed{A^t=\left[\begin{array}{cc}-1&1\\-4&-2\end{array}\right] }$

c)

$\det{A}=\left|\begin{array}{cc}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right| \ \therefore\ \det{A}=\left|\begin{array}{cc}-1&-4\\1&-2\end{array}\right|\ \therefore$

$\det{A}=-1(-2)-(-4)(1)=2+4\ \therefore\ \boxed{\det{A}=6}$