Math 2250
Homework 3
Due Wednesday, February 18
Leave in the box on my office door (Ross 313).
(1) If A=1−2
−2 5 , and AB =−1 2 −1
6−9 3 , determine the first and second
columns of B.
(2) If A=1−2 1
−102,B=5 2 −1
6 8 3 , and C=3 2
1 2 , find the following,
if possible.
(a) 2A−3B
(b) CB
(c) BC
(d) AT
(3) Suppose that a,B, and Care invertible n×nmatrices. Show that AB C is also
invertible by constructing a matrix Dsuch that (ABC )D=D(ABC ) = In.
(4) Find the inverse of the matrix A=
2 1 −1
021
−2 1 1
(5) Use Theorem 8 from Section 2.3 to answer the following:
(a) If the columns of a 7 ×7 matrix Dare linearly independent, what can you say
about solutions of Dx=b.
(b) If the equation Gx=yhas more than one solution for some yin Rn, can the
columns of Gspan Rn?
(c) If the equation Hx=cis inconsistent for some cin Rn, what can you say
about the equation Hx= 0?
(d) If the n×nmatrix Kcannot be row reduced to In, what can you say about
the columns of K?
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