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 , 0i
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 = h2 , 1 , 0 , 1i
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
0
