Elementary row operations with matrix multiplication
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Exercise:
Let Ain M_ntimes pK. abcliste abc Multiplying A from the left with Q_ijalpha results in the row operation R_i+alpha R_j longrightarrow R_i: A xrightarrowR_i+alpha R_j rightarrow R_i Q_ijalpha A abc Multiplying A from the left with P_ij results in the row operation R_i longleftrightarrow R_j: A xrightarrowR_ilongleftrightarrow R_j P_ij A abc Multiplying A from the left with S_ialpha results in the row operation alpha R_i longrightarrow R_i: A xrightarrowalpha R_ilongrightarrow R_i S_ialpha A abcliste The same holds also for elementary column operations: abcliste abc Multiplying A from the right with Q_ijalpha results in the col operation c_j+alpha c_i longrightarrow c_j: A xrightarrowc_j+alpha c_i longrightarrow c_j A Q_ijalpha abc Multiplying A from the right with P_ij results in the col operation c_i longleftrightarrow c_j: A xrightarrowc_ilongleftrightarrow c_j A P_ij abc Multiplying A from the right with S_ialpha results in the col operation alpha c_i longrightarrow c_i: A xrightarrowalpha c_ilongrightarrow c_i A S_ialpha abcliste
Solution:
Proof. a to c can be proven by direct calculation a' to c' can be deduced by passing to the transposed matrix e.g. A Q_ijalpha^TQ_ijalpha^T A^T Q_jialpha A^T textmatrix obtained from A^T after applying to it R_j+alpha R_i longrightarrow R_j textmatrix obtained from A after c_j+alpha c_ilongrightarrow c_j^T Longrightarrow A Q_ijalphatextmatrix obtained from A after c_j+alpha c_i longrightarrow c_j b and c can be shown in a similar way.
Let Ain M_ntimes pK. abcliste abc Multiplying A from the left with Q_ijalpha results in the row operation R_i+alpha R_j longrightarrow R_i: A xrightarrowR_i+alpha R_j rightarrow R_i Q_ijalpha A abc Multiplying A from the left with P_ij results in the row operation R_i longleftrightarrow R_j: A xrightarrowR_ilongleftrightarrow R_j P_ij A abc Multiplying A from the left with S_ialpha results in the row operation alpha R_i longrightarrow R_i: A xrightarrowalpha R_ilongrightarrow R_i S_ialpha A abcliste The same holds also for elementary column operations: abcliste abc Multiplying A from the right with Q_ijalpha results in the col operation c_j+alpha c_i longrightarrow c_j: A xrightarrowc_j+alpha c_i longrightarrow c_j A Q_ijalpha abc Multiplying A from the right with P_ij results in the col operation c_i longleftrightarrow c_j: A xrightarrowc_ilongleftrightarrow c_j A P_ij abc Multiplying A from the right with S_ialpha results in the col operation alpha c_i longrightarrow c_i: A xrightarrowalpha c_ilongrightarrow c_i A S_ialpha abcliste
Solution:
Proof. a to c can be proven by direct calculation a' to c' can be deduced by passing to the transposed matrix e.g. A Q_ijalpha^TQ_ijalpha^T A^T Q_jialpha A^T textmatrix obtained from A^T after applying to it R_j+alpha R_i longrightarrow R_j textmatrix obtained from A after c_j+alpha c_ilongrightarrow c_j^T Longrightarrow A Q_ijalphatextmatrix obtained from A after c_j+alpha c_i longrightarrow c_j b and c can be shown in a similar way.