Problem 2. Prove the following three useful matrix identities where the matrices are of compatible dimensions. (a) Prove the matrix identity (I + MN)' =I + M(I ± NM)' N HINT: One way to show that P = Q- is to establish PQ = I and %3D QP = I. REMARKS: M and N need not be square, but all the products and additions are of com- patible dimensioned matrices. In particular, (I+ MN)-' and (I+NM)-' may be matrices of different sizes. (b) Uuse (a) to show that MN(I + MN)-' = M(I+NM)-'N = (I+ MN)-'MN. c) Show that |I + MN| = |I + NM|. HINT: Let M and N be m x n and n x m matrices, respectively. Define [Im -M] A = Im Im B = C = Im M] c- -N In Show that |A||C| = |B||C| Then show and expand |AC| = |BC|.

Problem 2. Prove the following three useful matrix identities where the matrices are of compatible dimensions. (a) Prove the matrix identity (I + MN)' =I + M(I ± NM)' N HINT: One way to show that P = Q- is to establish PQ = I and %3D QP = I. REMARKS: M and N need not be square, but all the products and additions are of com- patible dimensioned matrices. In particular, (I+ MN)-' and (I+NM)-' may be matrices of different sizes. (b) Uuse (a) to show that MN(I + MN)-' = M(I+NM)-'N = (I+ MN)-'MN. c) Show that |I + MN| = |I + NM|. HINT: Let M and N be m x n and n x m matrices, respectively. Define [Im -M] A = Im Im B = C = Im M] c- -N In Show that |A||C| = |B||C| Then show and expand |AC| = |BC|.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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