Does there exist a \(3 \times 2\) complex matrix \(A\) such that \(A^t A = I_2\) and \(A A^t = I_3\)? If so, give an example; if not, prove it. Here \(I_k\) is the \(k \times k\) identity matrix.
<1>2. For every \(3\times2\) matrix \(A\), \[ \operatorname{rank}(AA^t)\le \operatorname{rank}(A)\le 2. \]
<1>3. Therefore \(AA^t\ne I_3\).
<1>4. Consequently there is no \(3\times2\) complex matrix satisfying both \(A^tA=I_2\) and \(AA^t=I_3\).
Solution
1. No such matrix exists.
The image of the linear map represented by \(AA^t\) is contained in the image of the linear map represented by \(A\), because \[ AA^t x=A(A^t x) \] for every \(x\in\mathbb C^3\). Hence \(\operatorname{rank}(AA^t)\le\operatorname{rank}(A)\). Since \(A\) has only two columns, \(\operatorname{rank}(A)\le2\).
The identity matrix \(I_3\) has rank \(3\), whereas 2 gives \(\operatorname{rank}(AA^t)\le2\).
The second required equality is already impossible by 3.