Math 2270 Practice Problems for Midterm 2
1. Practice Exam
1) Prove or disprove the following statement: matrix multiplication
is commutative.
2) Let A=
1−4 2
0−3 1
100
.Find A−1, using any method you deem
appropriate.
3) Let Abe a square matrix. State five conditions (not including the
definition) that are equivalent to the statement ‘Ais invertible’.
4) Let A=
3−1 2
−3−2 10
9−5 6
.Find an LU decomposition of A. (The
matrix Lwill be unit lower triangular.)
5) Prove or disprove: The determinant of a square matrix is the
product of the entries on the lead diagonal.
6) Consider the following statement: “If Ais a square matrix that is
row equivalent to U, and upper triangular matrix, then det A= det U”.
Explain why this statement is false, and give an additional condition
that will make the statement true.
7) Let Vbe a vector space containing at least two points. Find two
distinct subspaces of V.
8) Let Vbe a non-zero vector space. Let W⊂Vbe a non-zero
subspace. Let Bbe a (finite) collection of vectors in W. Give necessary
and sufficient conditions for Bto be a basis for W.
2. Additional Problems
9) Show that matrix addition is associative.
10) Suppose that Aand Bare invertible matrices of the same order.
Does it follow that AB is invertible?
11) Let Aand Bbe square matrices of the same order. Suppose AB
is invertible. Show that Ais invertible.
1
