Linear algebra and matrices M3P9

Question Sheet 3

1. Let A = (a_{ij}) be a strictly upper triangular matrix of size n × n, that is, a_{ij} = 0 if i ≥ j. Prove that A^{n} = 0.

2. Let V be the linear space of polynomials in one variable x of degrees less than n. Find the matrix of the linear transformation f(x) ↦ f′(x) (differentiation) in the basis 1, x, x^{2}/2, …, x^{n−1}/(n − 1)!. Find the matrix of f(x) ↦ af(x) + bf′′(x).

3. Let A and B be square matrices. Find the minimal polynomial of the block matrix

( \begin{matrix}A & 0 \\ 0 & B \end{matrix} )

4. Let A be a non-degenerate square matrix such that every row of A contains exactly one non-zero element, which is 1. Show that every eigenvalue of A is a root of 1 (that is, a complex number z such that z^{n} = 1 for some n). (Hint: Try to show that A^{n} = Id for some n.)

5. Prove that A from the previous exercise is diagonalizable.