 Solution

Each plane rotation satisfies $R_j^T{R}_j=I$. Hence $U=R_3{R}_2{R}_1$ is an orthogonal matrix and $D=UA$ is an upper triangular matrix. Multiplication by $U^T$ gives the QR decompositionThe zero-pair cases already handled make this construction valid for every real $3\times 3$ matrix, including rank-deficient matrices. Orthogonality follows from the product identity, not from any invertibility assumption on $A$.