Determinant and inverse matrices
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Exercise:
A is invertible iff ad-bcneq . Moreover if A is invertible then A^-fracad-bc pmatrix d & -b -c & a pmatrix. The number ad-bc is textdetA the determinant of A.
Solution:
Proof. A textis not invertible &iff textrankA &iff textits columns are linearly depent &iff leftarrayc a c arrayright lambda leftarrayc b d arrayright textor leftarrayc b d arrayright tau leftarrayc a c arrayright textIf leftarrayc a c arrayright lambda leftarrayc b d arrayright &Longrightarrow alambda b clambda d Longrightarrow ad-bc lambda bd- lambda bd textIf leftarrayc b d arrayright tau leftarrayc a c arrayright &Longrightarrow btau a dtau c Longrightarrow ad-bc tau ac-tau ac We have proven that A is not invertible Longrightarrow ad-bc. Conversely ase ad-bc. textIf a&Longrightarrow bc textso either b textor c &Longrightarrow texteither A pmatrix & c & d pmatrix textor A pmatrix & b & d pmatrix. and in both cases A is not invertible. textSo ase aneq &Longrightarrow dfracbca &Longrightarrow A pmatrix a & b c & fracbca pmatrix &Longrightarrow textthe second col of A is a multiple of the first col. &Longrightarrow textindeed leftarrayc b fracbca arrayright fracba leftarrayc a c arrayright &Longrightarrow textthe columns of A are linearly depent &Longrightarrow A textis not invertible. A is not invertible iff ad-bc. The formula for A^- direct calculation. fracad-bc pmatrix d & -b -c & a pmatrix pmatrix a & b c & d pmatrix fracad-bc pmatrix ad-bc & bd-bd -ac+ac & -bc+ad pmatrix fracad-bc pmatrix ad-bc & & -bc+ad pmatrix pmatrix & & pmatrix
A is invertible iff ad-bcneq . Moreover if A is invertible then A^-fracad-bc pmatrix d & -b -c & a pmatrix. The number ad-bc is textdetA the determinant of A.
Solution:
Proof. A textis not invertible &iff textrankA &iff textits columns are linearly depent &iff leftarrayc a c arrayright lambda leftarrayc b d arrayright textor leftarrayc b d arrayright tau leftarrayc a c arrayright textIf leftarrayc a c arrayright lambda leftarrayc b d arrayright &Longrightarrow alambda b clambda d Longrightarrow ad-bc lambda bd- lambda bd textIf leftarrayc b d arrayright tau leftarrayc a c arrayright &Longrightarrow btau a dtau c Longrightarrow ad-bc tau ac-tau ac We have proven that A is not invertible Longrightarrow ad-bc. Conversely ase ad-bc. textIf a&Longrightarrow bc textso either b textor c &Longrightarrow texteither A pmatrix & c & d pmatrix textor A pmatrix & b & d pmatrix. and in both cases A is not invertible. textSo ase aneq &Longrightarrow dfracbca &Longrightarrow A pmatrix a & b c & fracbca pmatrix &Longrightarrow textthe second col of A is a multiple of the first col. &Longrightarrow textindeed leftarrayc b fracbca arrayright fracba leftarrayc a c arrayright &Longrightarrow textthe columns of A are linearly depent &Longrightarrow A textis not invertible. A is not invertible iff ad-bc. The formula for A^- direct calculation. fracad-bc pmatrix d & -b -c & a pmatrix pmatrix a & b c & d pmatrix fracad-bc pmatrix ad-bc & bd-bd -ac+ac & -bc+ad pmatrix fracad-bc pmatrix ad-bc & & -bc+ad pmatrix pmatrix & & pmatrix