IMC / 1995 / Problems / Day 1, P1

IMC 1995 · Day 1 · P1

Let $X$ be a nonsingular matrix with columns $X_1, X_2, \dots, X_n$ . Let $Y$ be a matrix with columns $X_2, X_3, \dots, X_n, 0$ . Show that the matrices $A = Y X^{-1}$ and $B = X^{-1} Y$ have rank $n-1$ and have only $0$ 's for eigenvalues.

Solution (official)

Let $J = (a_{ij})$ be the $n \times n$ matrix where $a_{ij} = 1$ if $i = j + 1$ and $a_{ij} = 0$ otherwise. The rank of $J$ is $n-1$ and its only eigenvalues are $0$ 's. Moreover $Y = XJ$ and $A = Y X^{-1} = X J X^{-1}$ , $B = X^{-1} Y = J$ . It follows that both $A$ and $B$ have rank $n-1$ with only $0$ 's for eigenvalues.

IMC 1995 · Day 1 · P1

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