Vargas Greene

10/22/2024 · Primary School

29. Let \[ A=\left[\begin{array}{rrrr}1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & -1 & 1 & 1\end{array}\right] \] We wish to determine all \( 4 \times 3 \) matrices \( X \) for which \( A X A=A \) Observe that \[ P A Q=\left[\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0\end{array}\right] \] where \( P \) and \( Q \) are the invertible matrices \[ P=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 2 & 1\end{array}\right] \text { and } Q=\left[\begin{array}{rrrr}1 & -1 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 1\end{array}\right] \] (a) Explain how you can use this to convert the problem of finding all \( X \) so that \( A X A=A \) to finding all \( 4 \times 3 \) matrices \( Y \) so that \( B Y B=B \), where \( B=P A Q \). Make sure to indicate how \( X \) and \( Y \) are related. (b) Find all \( Y \) so that \( B Y B=B \). (c) Use your answer in (b) to find all \( X \) so that \( A X A=A \). (d) Give 2 explicit \( X \) with \( A X A=A \).