# MATH 300 University of South Florida Rank and Nullity Theorem for Matrices Worksheet

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Zhenghao Chen
Final
1. For each of the following answer T if the statement is always true and F otherwise. In each
case A, B ∈ Rn×n .
(a) If A and B are row equivalent then det(A) = det(B)
(b) If Nul(A) = {~0} then Col(A) = Rn .
 
1

(c) If n = 3 and A 3 = ~0 then rank(A) ≤ 2.
2
(d) det(AB) = det(A) det(B)
(e) If B is created by swapping two columns of A the det(A) = − det(B)
(f) If A and B are row equivalent then Col(A) = Col(B)
(g) If Col(A) = .Rn then Nul(A) = {~0}.
(h) If A has an eigenbasis for Rn then A is invertible.
2. Let:

−4
3

0
A=
0

0
6

0 0 0 0
1 0 0 0

0 −3 0 1

0 −2 1 0

0 −2 0 0
2 0 0 0

−12
 7 

 −2 
~b = 

 1 

 −4 
14
(a) Find all solutions to A~x = ~b.
(b) Are the columns of A linearly independent?
(c) Do the columns of A span R6 ?
3. Consider:

−3 6
3 0 0
3
 2 −4 −2 1 0 −1

 6 −12 −6 3 −2 7 

A=

−3
6
3
0
−1
8

 6 −12 −6 0 −2 4 
8 −16 −8 1 0 −7
which it row equivalent to:

1 −2 −1 0 0 −1
0 0
0 1 0 1

0 0

0
0
1
−5

rref(A) = 
0 0
0 0 0 0

0 0
0 0 0 0
0 0
0 0 0 0
(a) Find a basis for Col(A).
(b) Find a basis for Nul(A).
(1)
(c) Find rank(A)
(d) Find nullity(A)
(e) Are the columns of A linearly independent?
(f) Do the columns of A span R6 ?
4. Let T : P3 → R2×2 be defined by:

a + 2b c
T (ax + bx + cx + d) =
3d
2c
3
2

You may assume T is a linear transformation without proof.
(a) Find a basis for ker(T ).
(b) Find dim(ker(T )).
(c) Find dim(Rg(T ))
(d) Find a basis for Rg(T )
(e) Is T onto?
5. Let:

−3 −4
A=
0 −1
.
(a) Find all the eigenvalues for A.
(b) For each eigenvalue, find a corresponding eigenvector.
(c) Is A diagonalizable?
6. Let:
and

2
0
0
A = 71 −17 30
38 −10 18
     
1
0 
 0
B = −3 , −1 , −2 .

−2
−3
−1
(a) Show that B is an eigenbasis with respect to A and find the corresponding eignenvalues?
Make sure to explain how you know it is a basis.
(b) Find D and P so that AP = P D.
7. Now that you have finished, scan this with an app like Genius Scan or even an actual scanner
to turn this into a pdf file (not a jpeg). Name the file “final.pdf”. Put this file in our shared
directory (not the test subdirectory). Do all this to get 2 points on the exam.
Page 2

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Tags:
Math Problems

Matrices

math worksheet

Row equivalent

Nullity Theorem

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