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Question Number 116633 by zakirullah last updated on 05/Oct/20

Answered by Olaf last updated on 05/Oct/20

$$\mathrm{If}\mathrm{B}\mathrm{is}\mathrm{a}\mathrm{inverse}\mathrm{matrix}\mathrm{of}\mathrm{A}$$$$\mathrm{then}\mathrm{B}.\mathrm{A}=\mathrm{A}.\mathrm{B}=\mathrm{I}\left(1\right)$$$$$$$$\mathrm{If}\mathrm{C}\mathrm{is}\mathrm{another}\mathrm{inverse}\mathrm{matrix}\mathrm{of}\mathrm{A}$$$$\mathrm{then}\mathrm{C}.\mathrm{A}=\mathrm{A}.\mathrm{C}=\mathrm{I}\left(2\right)$$$$$$$$\left(1\right)\mathrm{and}\left(2\right):$$$$\mathrm{I}=\mathrm{AB}=\mathrm{AC}$$$$\Rightarrow \mathrm{AB}-\mathrm{AC}=\stackrel{\sim }{0}\left(\mathrm{the}\mathrm{null}\mathrm{matrix}\right)$$$$\Rightarrow \mathrm{A}(\mathrm{B}-\mathrm{C})=\stackrel{\sim }{0}$$$$\Rightarrow \mathrm{A}=\stackrel{\sim }{0}\mathrm{impossible}\mathrm{because}$$$$\mathrm{A}\mathrm{is}\mathrm{inversible}.$$$$\mathrm{Or}\mathrm{B}-\mathrm{C}=\stackrel{\sim }{0}\Rightarrow \mathrm{B}=\mathrm{C}$$$$\mathrm{impossible}\mathrm{because}\mathrm{B}\mathrm{is}\mathrm{different}$$$$\mathrm{of}\mathrm{C}.\mathrm{Contradiction}!$$$$$$$$\mathrm{So}\mathrm{the}\mathrm{initial}\mathrm{hypothesis}\mathrm{is}\mathrm{false}.$$$$$$

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