MATH 253 Fall 2015
Matrix Algebra
Test 2 Info and Review Sheet
Test 2 is Thursday, November 19 in class (10:15-11:05). It covers 2.1-2.3, 3.1-3.5. All questions
are written answer, the total test is out of 20. You can use a calculator. The best way to
study is to do practice problems- so the sample questions listed on the website. The book
also has really good chapter summaries at the end of each chapter. You can use a
calculator, but be sure to explain your steps.
The main things you need to know:
 Vectors in Rm: length, scalar multiplication, sum, difference, dot product, orthogonal
 Line segment between two vectors, planes/hyperplanes
 Systems of linear equations: consistent and inconsistent
 Different ways to solve:
1. Using elementary operations on the equations
2. Using augmented matrix (Gauss-Jordan or Gaussian elimination)
3. Method of Inverses
4. LU Decomposition
 There will be at least one question where you have to show steps in doing a row
reduction
 Matrices: basic terminology (rows, columns, entries, dimensions), operations (sum,
difference, scalar multiplication, negative, transpose, multiplication, powers)
 Inverse of a matrix, when is a matrix invertible or non-invertible
 Every system of linear equations has no solution, exactly one solution, or infinitely many
solutions; homogeneous systems have at least the trivial solution
 Rank of a matrix and how it can be used to determine how many solutions there are
 Theorem 5 of section 3.3 is very powerful; you should be able to prove any part of it
 Using particular solutions to find general solutions (Theorem 6 of 3.3)
 Know how to use LU decomposition, you don’t need to know the algorithm to find the
LU decomposition (it would be given to you)
 Elementary matrices, expressing a step in row reduction in terms of premultiplication
by an elementary matrix
 Inverse of products (Theorem 2 of Section 3.5): should be able to prove and use
Theorem Info:
In general you should be familiar with theorems we saw in class in that you can apply them
to specific questions.