Exercises for practice lesson 6
(1) Find left side and right side inverses for the matrix
1 −1
2 1


1 2 3
0 1 4
5 6 0
(2) Find f (A) if f (x) = x2 + 2x − x−1 and


1 1 1
A = −1 0 1
3 5 8
(3) Find non-singular matrices P and Q, such that




1
3
0
1
1 0 0 0
P · −2 −6 −1 −4 · Q = 0 1 0 0
1
3
1
3
0 0 0 0
(4) Find bases such that this matrix represents the identity map with respect to
those bases.


3 1 4
2 −1 1
0 0 4
(5) Find the change of basis matrix for bases from B to D, where
     
     
1
1
1
1
0
0











B = h 1 , 2 , 3 i; D = h 2 , 1 , 0i
0
3
5
1
1
1
1
2
Home work 6
(1) Find left side and right side inverses for the matrix


1
1
1
0
−1 0
1 −1


3
5
8
0
0 −1 −1 4
(2) Find the change of basis matrix for bases from B to D, where
       
       
1
1
1
1
0
0
0
1
1 2 3 −1
1 0 0 0
      
       
B = h
0 , 3 , 5 ,  1 i; D = h2 , 1 , 0 , 1i
0
0
0
1
1
1
1
2
(3) Find non-singular matrices P and Q, such that




1 −1 0 1
3 2 3 1
−1 1 2 0


 · Q =  5 5 4 0
P ·
3
19 16 15 2
2 1 0
3
2 3 1
3 2 3 1
(4) Find f (A) if f (x) = 2x2 − 3x + x−1 and


2 −3 1 −3
−1 5 −2 5 

A=
 0 −2 1 −2
0
0
0
1