Respostas
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Olá
A = a² + b - c²
Lembrando que multiplicação de matrizes é dada por linhas vezes colunas.
![\displaystyle\mathsf{ a=\left[\begin{array}{ccc}2&1\\-3&4\\\end{array}\right] }\\\\\\\boxed{\mathsf{a^2=a\cdot a}}\\\\\\\mathsf{ a^2=\left[\begin{array}{ccc}2&1\\-3&4\\\end{array}\right] \cdot\left[\begin{array}{ccc}2&1\\-3&4\\\end{array}\right] }\\\\\\\mathsf{=\left[\begin{array}{ccc}(2\cdot2)+(1\cdot(-3))\qquad&(2\cdot1)+(1\cdot4)\\(-3\cdot2)+(4\cdot(-3))\qquad&(-3\cdot1)+(4\cdot4)\\\end{array}\right] }}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cmathsf%7B++a%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26amp%3B1%5C%5C-3%26amp%3B4%5C%5C%5Cend%7Barray%7D%5Cright%5D+%7D%5C%5C%5C%5C%5C%5C%5Cboxed%7B%5Cmathsf%7Ba%5E2%3Da%5Ccdot+a%7D%7D%5C%5C%5C%5C%5C%5C%5Cmathsf%7B++a%5E2%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26amp%3B1%5C%5C-3%26amp%3B4%5C%5C%5Cend%7Barray%7D%5Cright%5D+%5Ccdot%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%26amp%3B1%5C%5C-3%26amp%3B4%5C%5C%5Cend%7Barray%7D%5Cright%5D+%7D%5C%5C%5C%5C%5C%5C%5Cmathsf%7B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%282%5Ccdot2%29%2B%281%5Ccdot%28-3%29%29%5Cqquad%26amp%3B%282%5Ccdot1%29%2B%281%5Ccdot4%29%5C%5C%28-3%5Ccdot2%29%2B%284%5Ccdot%28-3%29%29%5Cqquad%26amp%3B%28-3%5Ccdot1%29%2B%284%5Ccdot4%29%5C%5C%5Cend%7Barray%7D%5Cright%5D+%7D%7D "\displaystyle\mathsf{ a=\left[\begin{array}{ccc}2&1\\-3&4\\\end{array}\right] }\\\\\\\boxed{\mathsf{a^2=a\cdot a}}\\\\\\\mathsf{ a^2=\left[\begin{array}{ccc}2&1\\-3&4\\\end{array}\right] \cdot\left[\begin{array}{ccc}2&1\\-3&4\\\end{array}\right] }\\\\\\\mathsf{=\left[\begin{array}{ccc}(2\cdot2)+(1\cdot(-3))\qquad&(2\cdot1)+(1\cdot4)\\(-3\cdot2)+(4\cdot(-3))\qquad&(-3\cdot1)+(4\cdot4)\\\end{array}\right] }}")
![\displaystyle\mathsf{=\left[\begin{array}{ccc}(4-3)\qquad&(2+4)\\(-6-12)\qquad&(-3+16)\\\end{array}\right] }}\\\\\\\boxed{\mathsf{a^2=\left[\begin{array}{ccc}1&6\\-18&13\\\end{array}\right] }}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cmathsf%7B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%284-3%29%5Cqquad%26amp%3B%282%2B4%29%5C%5C%28-6-12%29%5Cqquad%26amp%3B%28-3%2B16%29%5C%5C%5Cend%7Barray%7D%5Cright%5D+%7D%7D%5C%5C%5C%5C%5C%5C%5Cboxed%7B%5Cmathsf%7Ba%5E2%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26amp%3B6%5C%5C-18%26amp%3B13%5C%5C%5Cend%7Barray%7D%5Cright%5D+%7D%7D%7D "\displaystyle\mathsf{=\left[\begin{array}{ccc}(4-3)\qquad&(2+4)\\(-6-12)\qquad&(-3+16)\\\end{array}\right] }}\\\\\\\boxed{\mathsf{a^2=\left[\begin{array}{ccc}1&6\\-18&13\\\end{array}\right] }}}")
O mesmo processo deve ser feito para obter c²
![\displaystyle\mathsf{ c=\left[\begin{array}{ccc}-1&-2\\5&3\\\end{array}\right] }\\\\\\\boxed{\mathsf{c^2=c\cdot c}}\\\\\\\mathsf{ c^2=\left[\begin{array}{ccc}-1&-2\\5&3\\\end{array}\right] \cdot\left[\begin{array}{ccc}-1&-2\\5&3\\\end{array}\right]}\\\\\\\mathsf{=\left[\begin{array}{ccc}(-1\cdot(-1))+(-2\cdot5)\qquad&(-1\cdot(-2))+(-2\cdot3)\\(5\cdot(-1))+(3\cdot5)\qquad&(5\cdot(-2))+(3\cdot3)\\\end{array}\right] }}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cmathsf%7B+c%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-1%26amp%3B-2%5C%5C5%26amp%3B3%5C%5C%5Cend%7Barray%7D%5Cright%5D+%7D%5C%5C%5C%5C%5C%5C%5Cboxed%7B%5Cmathsf%7Bc%5E2%3Dc%5Ccdot+c%7D%7D%5C%5C%5C%5C%5C%5C%5Cmathsf%7B+c%5E2%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-1%26amp%3B-2%5C%5C5%26amp%3B3%5C%5C%5Cend%7Barray%7D%5Cright%5D+%5Ccdot%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-1%26amp%3B-2%5C%5C5%26amp%3B3%5C%5C%5Cend%7Barray%7D%5Cright%5D%7D%5C%5C%5C%5C%5C%5C%5Cmathsf%7B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%28-1%5Ccdot%28-1%29%29%2B%28-2%5Ccdot5%29%5Cqquad%26amp%3B%28-1%5Ccdot%28-2%29%29%2B%28-2%5Ccdot3%29%5C%5C%285%5Ccdot%28-1%29%29%2B%283%5Ccdot5%29%5Cqquad%26amp%3B%285%5Ccdot%28-2%29%29%2B%283%5Ccdot3%29%5C%5C%5Cend%7Barray%7D%5Cright%5D+%7D%7D "\displaystyle\mathsf{ c=\left[\begin{array}{ccc}-1&-2\\5&3\\\end{array}\right] }\\\\\\\boxed{\mathsf{c^2=c\cdot c}}\\\\\\\mathsf{ c^2=\left[\begin{array}{ccc}-1&-2\\5&3\\\end{array}\right] \cdot\left[\begin{array}{ccc}-1&-2\\5&3\\\end{array}\right]}\\\\\\\mathsf{=\left[\begin{array}{ccc}(-1\cdot(-1))+(-2\cdot5)\qquad&(-1\cdot(-2))+(-2\cdot3)\\(5\cdot(-1))+(3\cdot5)\qquad&(5\cdot(-2))+(3\cdot3)\\\end{array}\right] }}")
![\displaystyle\mathsf{=\left[\begin{array}{ccc}(1-10)\qquad&(2-6)\\(-5+15)\qquad&(-10+9)\\\end{array}\right] }}\\\\\\\boxed{\mathsf{c^2=\left[\begin{array}{ccc}-9&-4\\10&-1\\\end{array}\right] }}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cmathsf%7B%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%281-10%29%5Cqquad%26amp%3B%282-6%29%5C%5C%28-5%2B15%29%5Cqquad%26amp%3B%28-10%2B9%29%5C%5C%5Cend%7Barray%7D%5Cright%5D+%7D%7D%5C%5C%5C%5C%5C%5C%5Cboxed%7B%5Cmathsf%7Bc%5E2%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-9%26amp%3B-4%5C%5C10%26amp%3B-1%5C%5C%5Cend%7Barray%7D%5Cright%5D+%7D%7D%7D "\displaystyle\mathsf{=\left[\begin{array}{ccc}(1-10)\qquad&(2-6)\\(-5+15)\qquad&(-10+9)\\\end{array}\right] }}\\\\\\\boxed{\mathsf{c^2=\left[\begin{array}{ccc}-9&-4\\10&-1\\\end{array}\right] }}}")
Como b não muda, Agora basta fazer a operação final:
A = a² + b - c²
![\displaystyle\mathsf{A =\left[\begin{array}{ccc}1&6\\-18&13\\\end{array}\right] +\left[\begin{array}{ccc}21&7\\-3&1\\\end{array}\right]-\left[\begin{array}{ccc}-9&-4\\10&-1\\\end{array}\right]}\\\\\\\\\mathsf{A=\left[\begin{array}{ccc}(1+21-(-9))\qquad&(6+7-(-4))\\(-18+(-3)-10)\qquad&(13+1-(-1))\\\end{array}\right] }\\\\\\\\\boxed{\boxed{\mathsf{A=\left[\begin{array}{ccc}31&17\\-31&15\\\end{array}\right] }}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cmathsf%7BA+%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%26amp%3B6%5C%5C-18%26amp%3B13%5C%5C%5Cend%7Barray%7D%5Cright%5D+%2B%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D21%26amp%3B7%5C%5C-3%26amp%3B1%5C%5C%5Cend%7Barray%7D%5Cright%5D-%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D-9%26amp%3B-4%5C%5C10%26amp%3B-1%5C%5C%5Cend%7Barray%7D%5Cright%5D%7D%5C%5C%5C%5C%5C%5C%5C%5C%5Cmathsf%7BA%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%281%2B21-%28-9%29%29%5Cqquad%26amp%3B%286%2B7-%28-4%29%29%5C%5C%28-18%2B%28-3%29-10%29%5Cqquad%26amp%3B%2813%2B1-%28-1%29%29%5C%5C%5Cend%7Barray%7D%5Cright%5D+%7D%5C%5C%5C%5C%5C%5C%5C%5C%5Cboxed%7B%5Cboxed%7B%5Cmathsf%7BA%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D31%26amp%3B17%5C%5C-31%26amp%3B15%5C%5C%5Cend%7Barray%7D%5Cright%5D+%7D%7D%7D "\displaystyle\mathsf{A =\left[\begin{array}{ccc}1&6\\-18&13\\\end{array}\right] +\left[\begin{array}{ccc}21&7\\-3&1\\\end{array}\right]-\left[\begin{array}{ccc}-9&-4\\10&-1\\\end{array}\right]}\\\\\\\\\mathsf{A=\left[\begin{array}{ccc}(1+21-(-9))\qquad&(6+7-(-4))\\(-18+(-3)-10)\qquad&(13+1-(-1))\\\end{array}\right] }\\\\\\\\\boxed{\boxed{\mathsf{A=\left[\begin{array}{ccc}31&17\\-31&15\\\end{array}\right] }}}")
A = a² + b - c²
Lembrando que multiplicação de matrizes é dada por linhas vezes colunas.
O mesmo processo deve ser feito para obter c²
Como b não muda, Agora basta fazer a operação final:
A = a² + b - c²
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