Math 3A
Summer 2025
Linear Algebra - Sections: 2.3 , 2.8, and 2.9
Problem 1.
(a) Find the inverse of the matrix
8
2
A=
.
−3 −2
(b) When is a 3 × 3 matrix invertible?
1
Problem 2.
Use matrix algebra to show that if A is invertible and D satisfies AD = I, then
D = A−1 .
Problem 3.
True or False: If A is an n × n matrix, then the equation Ax = b has at least one
solution for each b ∈ Rn .
2
Problem 4.
True or False: If Ax = b has at least one solution for each b ∈ Rn , then the solution
is unique for each b.
Problem 5.
Suppose the columns of A = [a1 , ..., ap ] are linearly independent. Explain why
{a1 , . . . , ap } is a basis for Col A.
3
Problem 6.
A matrix A and an echelon form of A are shown below. Find a basis for Col A and
a basis for Nul A.

 

1 0 −5 −2 −1
1 10 11 0 5
−1 10 21 4


7
 ∼ 0 5 8 0 −2
A=
−2 10 26 7 13  0 0 0 1 5 
2 5 −2 −5 −9
0 0 0 0 0
Find a basis for Col A.
Find a basis for Nul A.
4
Problem 7.
True or False: The null space of an m × n matrix is a subspace of Rn .
Problem 8.
True or False: The columns of an invertible n × n matrix form a basis for Rn .
5
Problem 9.
True or false: A subspace of Rn is any set H such that:
(i) 0 ∈ H
(ii) u, v ∈ H ⇒ u + v ∈ H
(iii) u ∈ H, c ∈ R ⇒ cu ∈ H.
Problem 10.
True or False: A subset H of Rn is a subspace if the zero vector is in H.
6