Question

1)Construa a matriz A = (aij)3x3, tal que aij = 3j - 2i. Mostre sua matriz transposta.
2)Construa a matriz B = (bij)3x2, tal que bij = i + 2j. Calcule o determinante.
3)Verifique e comprove se são côngruos os arcos a seguir: 14π/3 rad e 25π/4 rad.
4)Sabendo que o comprimento de uma circunferência é de 54cm, calcule o seu diâmetro.

Answer 1

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$\rm{1)}~\sf{a_{11}=3\cdot1-2\cdot1=1~~a_{12}=3\cdot2-2\cdot1=4~~a_{13}=3\cdot3-2\cdot1=7}\\\sf{a_{21}=3\cdot1-2\cdot2=-1~~a_{22}=3\cdot2-2\cdot2=2~~a_{23}=3\cdot3-2\cdot2=5}\\\sf{a_{31}=3\cdot1-2\cdot3=-3~~a_{32}=3\cdot2-2\cdot3=0~~a_{33}=3\cdot3-2\cdot3=3}\\\tt{A}=\begin{bmatrix}\sf{1}&\sf{4}&\sf{7}\\\sf{-1}&\sf{2}&\sf{5}\\\sf{-3}&\sf{0}&\sf{3}\end{bmatrix}\\\tt{A^T}=\begin{bmatrix}\sf{1}&\sf{-1}&\sf{3}\\\sf{4}&\sf{2}&\sf{0}\\\sf{7}&\sf{5}&\sf{3}\end{bmatrix}$

$\rm{2)}~\sf{a_ {11}=1+2\cdot1=3~~a_{12}=1+2\cdot2=5}\\\sf{a_{21}=2+2\cdot1 =4~~a_{22}=2+2\cdot2=6}\\\sf{a_{31}=3+2\cdot1=5~~a_{32}=3+2\cdot2=7}\\\sf{A}=\begin{bmatrix}\sf{3}&\sf{5}\\\sf{4}&\sf{6}\\\sf{5}&\sf{7}\end{bmatrix}\\\sf{S\acute{o}~existe~determinante~de~matriz~quadrada}$

$\rm{3)}\\\sf{\dfrac{14\pi}{3}=4\pi+\dfrac{2\pi}{3}=2\cdot2\pi+\boxed{\dfrac{2\pi}{3}}}\\\sf{\dfrac{25\pi}{4}=6\pi+\dfrac{\pi}{4}=3\cdot2\pi+\boxed{\dfrac{\pi}{4}}}\\\sf{como~\dfrac{2\pi}{3}\ne\dfrac{\pi}{4}~os~arcos~n\tilde{a}o~s\tilde{a}o~c\hat{o}ngruos}$

$\rm{4)}\\\sf{C=D\cdot\pi\implies D=\dfrac{C}{\pi}}\\\sf{D=\dfrac{54}{3,14}}\\\boxed{\boxed{\boxed{\boxed{\sf{D=17,19~cm}}}}}$