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Exercise:
Let Ain M_ntimes nK with textdetAneq . Let bin K^n. Then the system Axb has a unique solution and it is giben by the formula x_jfractextdetAjbtextdetAquad forall leq jleq n where Ajb is the matrix obtained from A by replacing col#j by b.

Solution:
Proof. We have seen linalg I that if A is invertible then Axb has a unique solution. We have seen above that textdetA x_jtextdetAjb Longrightarrow x_jfractextdetAjbtextdetA.
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Exercise:
Let Ain M_ntimes nK with textdetAneq . Let bin K^n. Then the system Axb has a unique solution and it is giben by the formula x_jfractextdetAjbtextdetAquad forall leq jleq n where Ajb is the matrix obtained from A by replacing col#j by b.

Solution:
Proof. We have seen linalg I that if A is invertible then Axb has a unique solution. We have seen above that textdetA x_jtextdetAjb Longrightarrow x_jfractextdetAjbtextdetA.
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Tags
determinant, eth, fs23, lineare algebra, matrices, proof
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Difficulty
(3, default)
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0 (default)
Language
ENG (English)
Type
Proof
Creator rk
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