Homework for §3.5
Solve the system of equations A~x = ~b for the given choices of matrix A and vector ~b by first
finding A−1 and then computing ~x = A−1~b.
3
−1 1
~
.
, b=
1. A =
3
2 1


 
1 0
4
3
~



2 −1 3
2 .
2. A =
, b=
0 2 −2
0
 


2
1 0 0 2
 6 
 0 0 1 1 
 ~  
3. A = 
 0 1 0 0 , b =  4 .
1
0 0 0 3
.........................................................................................
4. Solve the matrix equation AX = B, where
2 1 −2
2 3
and B =
A=
3 4 6
−1 9
by computing X = A−1 B.
5. Suppose that A is an n × n matrix such that A~x = ~x for every n × 1 vector ~x. Explain why
it must be that A = I. (Hint: Saying “because the identity is the only matrix that has this
property” is not an explanation.)