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<span class="iDeviceTitle">Ejercicio selectivad matrices</span>
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Consideremos la matriz
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<img src="http://latex.codecogs.com/gif.latex?A=\begin{pmatrix}%200&%203&%204\\%201&%20-4&%20-5\\%20-1&%203&%204%20\end{pmatrix}" title="A=\begin{pmatrix} 0& 3& 4\\ 1& -4& -5\\ -1& 3& 4 \end{pmatrix}" /><br />
a) Siendo I la matriz identidad 3 x 3 y O la matriz nula 3x3, probar que <img src="http://latex.codecogs.com/gif.latex?A^{3}" title="A^{3}" />+I=0
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b) Calcula <img src="http://latex.codecogs.com/gif.latex?A^{10}" title="A^{10}" />
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a)
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<img src="http://latex.codecogs.com/gif.latex?A^{2}=A\cdot%20A=\begin{pmatrix}%200&%203&%204\\%201&%20-4&%20-5\\%20-1&%203&%204%20\end{pmatrix}\cdot%20\begin{pmatrix}%200&%203&%204\\%201&%20-4&%20-5\\%20-1&%203&%204%20\end{pmatrix}=\begin{pmatrix}%20-1&%200&%201\\%201&%204&%204\\%20-1&%20-3&%20-3%20\end{pmatrix}" title="A^{2}=A\cdot A=\begin{pmatrix} 0& 3& 4\\ 1& -4& -5\\ -1& 3& 4 \end{pmatrix}\cdot \begin{pmatrix} 0& 3& 4\\ 1& -4& -5\\ -1& 3& 4 \end{pmatrix}=\begin{pmatrix} -1& 0& 1\\ 1& 4& 4\\ -1& -3& -3 \end{pmatrix}" />
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<img src="http://latex.codecogs.com/gif.latex?A^{3}=A^{2}\cdot%20A=\begin{pmatrix}%20-1&%200&%201\\%201&%204&%204\\%20-1&%20-3&%20-3%20\end{pmatrix}\cdot%20\begin{pmatrix}%200&%203&%204\\%201&%20-4&%20-5\\%20-1&%203&%204%20\end{pmatrix}=\begin{pmatrix}%20-1&%200&%200\\%200&%20-1&%200\\%200&%200&%20-1%20\end{pmatrix}" title="A^{3}=A^{2}\cdot A=\begin{pmatrix} -1& 0& 1\\ 1& 4& 4\\ -1& -3& -3 \end{pmatrix}\cdot \begin{pmatrix} 0& 3& 4\\ 1& -4& -5\\ -1& 3& 4 \end{pmatrix}=\begin{pmatrix} -1& 0& 0\\ 0& -1& 0\\ 0& 0& -1 \end{pmatrix}" /><br />
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<img src="http://latex.codecogs.com/gif.latex?A^{3}+I=\begin{pmatrix}%200&%200&%200\\%200&%200&%200\\%200&%200&%200%20\end{pmatrix}" title="A^{3}+I=\begin{pmatrix} 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0 \end{pmatrix}" /><br />
b)
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<img src="http://latex.codecogs.com/gif.latex?A%5E%7B10%7D=A%5E%7B3%7D%5Ccdot%20A%5E%7B3%7D%5Ccdot%20A%5E%7B3%7D%5Ccdot%20A=%28-I%29%5Ccdot%28-I%29%5Ccdot%20%28-I%29%5Ccdot%20A=-A=%5Cbegin%7Bpmatrix%7D%200&%20-3&%20-4%5C%5C%20-1&%204&%205%5C%5C%201&%20-3&%20-4%20%5Cend%7Bpmatrix%7D" title="A^{10}=A^{3}\cdot A^{3}\cdot A^{3}\cdot A=(-I)\cdot(-I)\cdot (-I)\cdot A=-A=\begin{pmatrix} 0& -3& -4\\ -1& 4& 5\\ 1& -3& -4 \end{pmatrix}" />
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<span class="iDeviceTitle">Ejercico selectividad matrices</span>
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Considerar las matrices
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<img src="http://latex.codecogs.com/gif.latex?A=\begin{pmatrix}%201&%200&%201\\%200&%201&%202%20\end{pmatrix}" title="A=\begin{pmatrix} 1& 0& 1\\ 0& 1& 2 \end{pmatrix}" /> <img src="http://latex.codecogs.com/gif.latex?B=\begin{pmatrix}%201&%200\\%200&%201\\%200&%200%20\end{pmatrix}" title="B=\begin{pmatrix} 1& 0\\ 0& 1\\ 0& 0 \end{pmatrix}" /> <img src="http://latex.codecogs.com/gif.latex?c=\begin{pmatrix}%201&%200\\%200&%202\\%201&%200%20\end{pmatrix}" title="c=\begin{pmatrix} 1& 0\\ 0& 2\\ 1& 0 \end{pmatrix}" />
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a) Calcular <img src="http://latex.codecogs.com/gif.latex?A\cdot%20B,%20A\cdot%20C,%20A^{t}\cdot%20B^{t},C^{t}\cdot%20A^{t}" title="A\cdot B, A\cdot C, A^{t}\cdot B^{t},C^{t}\cdot A^{t}" /> siendo <img src="http://latex.codecogs.com/gif.latex?A^{t},%20B^{t}" title="A^{t}, B^{t}" /> y <img src="http://latex.codecogs.com/gif.latex?C%5E%7Bt%20%7D" title="C^{t }" /> las matrices traspuestas de A,B Y C respectivamente.
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<img src="http://latex.codecogs.com/gif.latex?A\cdot%20B=\begin{pmatrix}%201&%200&%201\\%200&%201&%202%20\end{pmatrix}\cdot%20\begin{pmatrix}%201&%200\\%200&%201\\%200&%200%20\end{pmatrix}=\begin{pmatrix}%201&%200\\%200&%201%20\end{pmatrix}=I" title="A\cdot B=\begin{pmatrix} 1& 0& 1\\ 0& 1& 2 \end{pmatrix}\cdot \begin{pmatrix} 1& 0\\ 0& 1\\ 0& 0 \end{pmatrix}=\begin{pmatrix} 1& 0\\ 0& 1 \end{pmatrix}=I" />
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<img src="http://latex.codecogs.com/gif.latex?A.C=\begin{pmatrix}%201&%200&%201\\%200&%201&%202%20\end{pmatrix}\cdot%20\begin{pmatrix}%201&%200\\%200&%202\\%201&%200%20\end{pmatrix}=\begin{pmatrix}%202&%200\\%202&%202%20\end{pmatrix}" title="A.C=\begin{pmatrix} 1& 0& 1\\ 0& 1& 2 \end{pmatrix}\cdot \begin{pmatrix} 1& 0\\ 0& 2\\ 1& 0 \end{pmatrix}=\begin{pmatrix} 2& 0\\ 2& 2 \end{pmatrix}" />
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<img src="http://latex.codecogs.com/gif.latex?A^{t}\cdot%20B^{t}=\begin{pmatrix}%201&%200\\%200&%201\\%201&%202%20\end{pmatrix}\cdot%20\begin{pmatrix}%201%20&%200&%200\\%200&%201&%200%20\end{pmatrix}=\begin{pmatrix}%201&%200&%200\\%200&%201&%200\\%201&%202&%200%20\end{pmatrix}" title="A^{t}\cdot B^{t}=\begin{pmatrix} 1& 0\\ 0& 1\\ 1& 2 \end{pmatrix}\cdot \begin{pmatrix} 1 & 0& 0\\ 0& 1& 0 \end{pmatrix}=\begin{pmatrix} 1& 0& 0\\ 0& 1& 0\\ 1& 2& 0 \end{pmatrix}" />
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<img src="http://latex.codecogs.com/gif.latex?C%5E%7Bt%7D%5Ccdot%20A%5E%7Bt%7D=%5Cbegin%7Bpmatrix%7D%201&%200&%201%5C%5C%200&%201&%202%20%5Cend%7Bpmatrix%7D%5Ccdot%20%5Cbegin%7Bpmatrix%7D%201&%200%5C%5C%200&%201%5C%5C%200&%200%20%5Cend%7Bpmatrix%7D=%5Cbegin%7Bpmatrix%7D%202&%202%5C%5C%200&%202%20%5Cend%7Bpmatrix%7D" title="C^{t}\cdot A^{t}=\begin{pmatrix} 1& 0& 1\\ 0& 1& 2 \end{pmatrix}\cdot \begin{pmatrix} 1& 0\\ 0& 1\\ 0& 0 \end{pmatrix}=\begin{pmatrix} 2& 2\\ 0& 2 \end{pmatrix}" />
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<span class="iDeviceTitle">Ejercicios selectividad matrices</span>
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Consideremos la matriz:
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<img src="http://latex.codecogs.com/gif.latex?A=\begin{pmatrix}%20a&%201\\%200&%20-a%20\end{pmatrix}" title="A=\begin{pmatrix} a& 1\\ 0& -a \end{pmatrix}" /><br />
a) Calcular el valor de a para que
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<img src="http://latex.codecogs.com/gif.latex?A%5E%7B2%7D-A=%5Cbegin%7Bpmatrix%7D%2012&%20-1%5C%5C%200&%2020%20%5Cend%7Bpmatrix%7D" title="A^{2}-A=\begin{pmatrix} 12& -1\\ 0& 20 \end{pmatrix}" />
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b) Existe algun valor de a para el cual la matriz A es simétrica. Razona la respuesta
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<img src="http://latex.codecogs.com/gif.latex?A=\begin{pmatrix}%20a&%201\\%200&%20-a%20\end{pmatrix}" title="A=\begin{pmatrix} a& 1\\ 0& -a \end{pmatrix}" />
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<img src="http://latex.codecogs.com/gif.latex?A^{2}=A\cdot%20A=\begin{pmatrix}%20a^{2}&%200\\%200&%20a^{2}%20\end{pmatrix}" title="A^{2}=A\cdot A=\begin{pmatrix} a^{2}& 0\\ 0& a^{2} \end{pmatrix}" />
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<img src="http://latex.codecogs.com/gif.latex?A^{2}-A=\begin{pmatrix}%2012&%20-1\\%200&%2020%20\end{pmatrix}=\begin{pmatrix}%20a^{2}&%200\\%200&%20a^{2}%20\end{pmatrix}-\begin{pmatrix}%20a&%201\\%200&%20-a%20\end{pmatrix}=\begin{pmatrix}%20a^{2}-a&%20-1\\%200&%20a^{2}+a%20\end{pmatrix}" title="A^{2}-A=\begin{pmatrix} 12& -1\\ 0& 20 \end{pmatrix}=\begin{pmatrix} a^{2}& 0\\ 0& a^{2} \end{pmatrix}-\begin{pmatrix} a& 1\\ 0& -a \end{pmatrix}=\begin{pmatrix} a^{2}-a& -1\\ 0& a^{2}+a \end{pmatrix}" />
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De donde <img src="http://latex.codecogs.com/gif.latex?a^{2}-a=12" title="a^{2}-a=12" /> y <img src="http://latex.codecogs.com/gif.latex?a^{2}+a=20" title="a^{2}+a=20" />. Sumando obtenemos que <img src="http://latex.codecogs.com/gif.latex?a=%5Cpm%204" title="a=\pm 4" />
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b) Sabemos que una matriz es simétrica si coincide con su traspuesta. En las matrices simétricas se tiene la propiedad de que sus elementos son simétricos respecto a la diagonal principal, cosa que no ocurre con la matriz <img src="http://latex.codecogs.com/gif.latex?A=%5Cbegin%7Bpmatrix%7D%20a&%201%5C%5C%200&%20-a%20%5Cend%7Bpmatrix%7D" title="A=\begin{pmatrix} a& 1\\ 0& -a \end{pmatrix}" />, pues uno es 0 y el otro es 1
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