Question

Fie matricea C= ( 3 -2 )
-1 2
Verificati relatia  C²-Tr(C)*C+det(C)*I2=O2 .

Answer 1

$\displaystyle \mathtt{C= \left(\begin{array}{ccc}\mathtt3&\mathtt{-2}\\\mathtt{-1}&\mathtt2\end{array}\right)~~~~~~~~~~~~~~~~~~~~~~~~~~C^2-Tr(C) \cdot C+det(C) \cdot I_2=O_2}$

$\displaystyle \mathtt{C^2=C \cdot C= \left(\begin{array}{ccc}\mathtt3&\mathtt{-2}\\\mathtt{-1}&\mathtt2\end{array}\right) \cdot \left(\begin{array}{ccc}\mathtt3&\mathtt{-2}\\\mathtt{-1}&\mathtt2\end{array}\right)=} \\ \\ \mathtt{= \left(\begin{array}{ccc}\mathtt{3 \cdot 3+(-2) \cdot (-1)}&\mathtt{3 \cdot (-2) +(-2)\cdot 2}\\\mathtt{(-1) \cdot 3+2 \cdot (-1)}&\mathtt{(-1) \cdot (-2)+2 \cdot 2}\end{array}\right)=}$

$\displaystyle \mathtt{ =\left(\begin{array}{ccc}\mathtt{9+2}&\mathtt{-6-4}\\\mathtt{-3-2}&\mathtt{2+4}\end{array}\right)= \left(\begin{array}{ccc}\mathtt{11}&\mathtt{-10}\\\mathtt{-5}&\mathtt6\end{array}\right)} }$

$\displaystyle \mathtt{Tr(C)=3+2=5} \\ \\ \mathtt{Tr(C) \cdot C=5 \cdot \left(\begin{array}{ccc}\mathtt{3}&\mathtt{-2}\\\mathtt{-1}&\mathtt2\end{array}\right)=\left(\begin{array}{ccc}\mathtt{5 \cdot 3}&\mathtt{5 \cdot (-2)}\\\mathtt{5 \cdot (-1)}&\mathtt5 \cdot 2\end{array}\right)=\left(\begin{array}{ccc}\mathtt{15}&\mathtt{-10}\\\mathtt{-5}&\mathtt10\end{array}\right)}$

$\displaystyle \mathtt{C^2-Tr(C) \cdot C=\left(\begin{array}{ccc}\mathtt{11}&\mathtt{-10}\\\mathtt{-5}&\mathtt6\end{array}\right) -\left(\begin{array}{ccc}\mathtt{15}&\mathtt{-10}\\\mathtt{-5}&\mathtt{10}\end{array}\right)=}\\ \\ \mathtt{\left(\begin{array}{ccc}\mathtt{11-15}&\mathtt{-10-(-10)}\\\mathtt{-5-(-5)}&\mathtt{6-10}\end{array}\right) =\left(\begin{array}{ccc}\mathtt{-4}&\mathtt{0}\\\mathtt{0}&\mathtt{-4}\end{array}\right)}$

$\displaystyle \mathtt{det(C)=\left|\begin{array}{ccc}\mathtt3&\mathtt{-2}\\\mathtt{-1}&\mathtt{2}\end{array}\right|=3 \cdot 2-(-2) \cdot (-1)=6-2=4}\\ \\ \mathtt{det(C) \cdot I_2=4 \cdot \left(\begin{array}{ccc}\mathtt{1}&\mathtt{0}\\\mathtt{0}&\mathtt{1}\end{array}\right)=\left(\begin{array}{ccc}\mathtt{4 \cdot 1}&\mathtt{4 \cdot 0}\\\mathtt{4 \cdot 0}&\mathtt{4 \cdot 1}\end{array}\right)=\left(\begin{array}{ccc}\mathtt{4}&\mathtt{0}\\\mathtt{0}&\mathtt{4}\end{array}\right)}$

$\displaystyle \mathtt{C^2-Tr(C) \cdot C+det(C) \cdot I_2=\left(\begin{array}{ccc}\mathtt{-4}&\mathtt{0}\\\mathtt{0}&\mathtt{-4}\end{array}\right) +\left(\begin{array}{ccc}\mathtt{4}&\mathtt{0}\\\mathtt{0}&\mathtt{4}\end{array}\right) =}\\ \\ \mathtt{=\left(\begin{array}{ccc}\mathtt{-4+4}&\mathtt{0+0}\\\mathtt{0+0}&\mathtt{-4+4}\end{array}\right)=\left(\begin{array}{ccc}\mathtt{0}&\mathtt{0}\\\mathtt{0}&\mathtt{0}\end{array}\right)=O_2}\\ \\ \mathtt{\Rightarrow C^2-Tr(C) \cdot C+det(C) \cdot I_2=O_2}$