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https://texercises.com/exercise/classical-adjoint-and-determinants/
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Exercise:
forall Ain M_ntimes nK forall jk we have _i^n A_ik C_ijdelta_jk textdetA. The matrix C_ij^T is called the classical adjo of A. textadjA&in M_ntimes nK textadjA_ijC_ji-^i+jtextdetAj|i The first formula above says that textadjA A textadjA I. Show that A textadjA textadjA I.

Solution:
Proof. A^Ti|jAj|i^T forall ij &Longrightarrow textadjA^T_ij -^i+jtextdetA^Tj|i -^i+jtextdetA^Ti|j^T -^i+jtextdetAi|j textadjA_ji textadjA^T_ij &Longrightarrow textadjA^T textadjA^T Apply the statement from the claim to A^T: textadjA^T A^T textdetA^T A^T textdetA^T I. Applying -^T to both sides of the last equality gets us to A textadjAtextdetA I
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Exercise:
forall Ain M_ntimes nK forall jk we have _i^n A_ik C_ijdelta_jk textdetA. The matrix C_ij^T is called the classical adjo of A. textadjA&in M_ntimes nK textadjA_ijC_ji-^i+jtextdetAj|i The first formula above says that textadjA A textadjA I. Show that A textadjA textadjA I.

Solution:
Proof. A^Ti|jAj|i^T forall ij &Longrightarrow textadjA^T_ij -^i+jtextdetA^Tj|i -^i+jtextdetA^Ti|j^T -^i+jtextdetAi|j textadjA_ji textadjA^T_ij &Longrightarrow textadjA^T textadjA^T Apply the statement from the claim to A^T: textadjA^T A^T textdetA^T A^T textdetA^T I. Applying -^T to both sides of the last equality gets us to A textadjAtextdetA I
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adjoint, determinant, eth, fs23, lineare algebra
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ENG (English)
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