1. For each matrix A below compute the characteristic polynomialχA(t) and do a direct matrix computation to verify that χA(A) =0.
(4 3
-1 1) Â Â (2 1 -1 0 3 0 0 -1 2) (3*3 matrix)
2.  For each 3*3 matrix and each eigenvalue belowconstruct a basis for the eigenspace Eλ.
A= (9 42 -30 -4 -25 20 -4 -28 23),λ = 1,3
A= (2 -27 18 0 -7 6 0 -9 8) , λ = −1,2
3. Construct a 2×2 matrix with eigenvectors(4 3) and (−3 −2)with eigen-values 2 and −3, respectively.
4. Let A be the 6*6 diagonal matrix below. For each eigenvalue,compute the multiplicity of λ as a root of the characteristicpolynomial and compare it to the dimension of the eigenspaceEλ.
(x 0 0 0 0 0 0 x 0 0 0 0 0 0 y 0 0 0 0 0 0 x 0 0 0 0 0 0 z 0 0 00 0 0 x)
5. Â Â Let A be an 3*3 upper triangular matrix with alldiagonal elements equal, such as (3 4 -2 0 3 12 0 0 3)
Prove that A is diagonalizable if and only if A is a scalartimes the identity matrix.
