Ejercicio 1. Determine if the following matrices are diagonalizable.
A=
1 0 0
0−1 1
0 0 −1
B=
1 0 1
0 1 −2
0 0 2
C=
5 1 −1
2 4 −2
1−1 3
Ejercicio 2. Determine the values of the parameters αand βfor which the following
matrix is diagonalizable:
A=
5 0 0
0−1α
3 0 β
Ejercicio 3. Given the matrix
A=
2−1 0
012
003
1) Show that Ais diagonalizable and determine its eigenvalues and eigenvectors.
2) Show that A=P DP −1.
3) Compute the eigenvalues of A−1.
Ejercicio 4. Given the matrix
A=
10000
01000
00100
00011
00011
1) Compute its characteristic equation.
2) Looking at its eigenvalues, determine if Ahas inverse.
3) Compute the eigenvector subspaces and study if Ais diagonalizable.
4) If it is diagonalizable, compute the diagonal form Dand the matrix Psuch that
A=P DP −1.
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