A square matrix A is called idempotent if A2 A. (The word idempotent comes from the Latin idem, meaning same, and potere, meaning to have power. Thus, something that is idempotent has the same power when squared.) (a) Find three idempotent 2 2 matrices. (b) Prove that the only invertible idempotent n n matrix is the identity matrix. 4

LECTURE 15: IMPROPER INTEGRALS (CONTINUED) Friday, February 12 Recall that an integral is improper if either one or more of the limits of integration are inﬁnite, or if the integral is over a ﬁnite interval ra,bs and the integrand has an inﬁnite discontinuity on that interval. We considered the former last time, in this lecture we’ll consider the latter. type 2: discontinuous integrands The following deﬁnite integrals are examples of type 2 improper integrals, » 1 1 » 3 1 . 0 x 0 x ▯ 1 Notice that the integrand of the ﬁrst integral has an inﬁnite dis- continuity at an endpoint of the interval r0,1s (at x ▯ 0), while the integrand of the second integral has an inﬁnite discontinuity in the interior of the interval r0,3s (at x ▯ 1). We handle improper integrals of this type in manner similar to the way we handled integrals over inﬁnite integrals. Deﬁnition 0.1. paqIf fpxq is continuous on ra,bq and discontinuous at b, then » » b t fpxqdx ▯ lim▯ fpxqdx a tÑb a provided this limit is a ﬁnite real number. pbq If fpxq is continuous on pa,bs and discontinuous at a, then » » b b fpxqdx ▯ lim fpxqdx