7. Let A be the square matrix given by 5 O O 0 5 1 0 A = 0 0 5 1 0 0 0 5 (a) Write down the characteristic polynomial PA(x) = det(A - x14) of the matrix A. (b) Show that A has only one eigenvalue, which we denote by A from now on. (c) Show that the eigenspace EA (A) is of dimension 1. Also name a basis for EA ()), where the only vector in the basis concerned is denoted by x1 from now on. (d) (Recall that by definition, EA (A) = N(A - XI4).) i. Name one vector x2 in R4 which belongs to N((A - XI4)2) but not N(A - XI4). ii. Name one vector x3 in R4 which belongs to N((A - XI4)3) but not N((A - XI4)2). iii. Name one vector x4 in R4 which belongs to N((A - Al4)4) but not N((A - AI4) 3). iv. Do X1, X2, X3, X4 constitute a basis for R4