Math 324: Assignment 5
Exercise 1. (a) Find the inverse of A=
1 2 −2
2 5 0
−3−6 5
(b) Write the matrix Aand A−1as products of elementary matrices. Check your answer using
MathCAD.
Exercise 2. Explain, in terms of row reducing, how you can tell whether a square matrix has
an inverse. Find a 3 by 3 matrix without an inverse, and illustrate your explaination.
Exercise 3. Suppose that A,Band Care nby nreal matrices such that
AB =In=CA
Prove that B=C.
Exercise 4. Suppose Ais invertible with inverse A−1, so AA−1=I=A−1A. If AC =I,
prove that C=A−1. In particular, inverses are unique.
Exercise 5. (a) Suppose E1is an elementary matrix obtained from Iby multiplying row 2 of
Iby −4. Describe E−1
1in terms of elemenatry row opertations on I.
(b) Suppose E2is an elementary matrix obtained by adding 4 times row 3 of Ito row 2.
Describe E−1
2in terms of elemenatry row opertations on I.
(c) Suppose E3is an elementary matrix obtained by switching rows 1 and 3 of I. Describe
E−1
3in terms of elemenatry row opertations on I.
(d) Verify your answers for 3 by 3 matrices.
Exercise 6. If A=EnEn−1. . . E3E2E1is a product elementary matrices. Express A−1as a
product of elementary matrices, and explain why your answer works.