91Ó°ÊÓ

Let \(A\) be a matrix with the property that columns \(1,3,5,7, \ldots\) are orthogonal to columns \(2,4,6,8, \ldots .\) In a reduced \(Q R\) factorization \(A=\hat{Q} \hat{R}\) what special structure does \(\hat{R}\) possess? You may assume that \(A\) has full rank

Let \(A\) be an \(m \times m\) matrix, and let \(a_{j}\) be its \(j\) th column. Give an algebraic proof of Hadamard's inequality \\[ |\operatorname{det} A| \leq \prod_{j=1}^{m}\left\|a_{j}\right\|_{2} \\] Also give a geometric interpretation of this result, making use of the fact that the determinant equals the volume of a parallelepiped.