Math 2530: Review Topics

General comments:

Remember the decimal rule: no decimals in the question means no decimals are allowed in your
answer!

Using a calculator is not an excuse for not showing any work. You will not receive credit for
steps you skip or do not write down. Use the examples from class as a guide for how much
work I expect.

If it can be done quickly, check your answers!

Know the equivalences of the Mega-Theorem of Death. Or else.

Section 1.1: Systems of Linear Equations

1. be able to identify a linear equation
2. consistent vs. inconsistent systems
3. expressing solutions using parameters
4. be able to transform a linear system into its augmented matrix
5. what are the three possibilities for the set of solutions to a linear system?

Section 1.2: Gaussian Elimination

1. we do not use substitution, addition or subtraction methods to solve systems of equations
2. make sure write to down what row operations you are performing in each step—if I have to guess,
that will cost you points!
3. row-echelon vs. reduced row-echelon form (which one is unique?), identifying leading ones
4. leading variables vs . free variables
5. Gaussian vs. Gauss-Jordan elimination, know the difference!
6. back substitution is a method we will rarely use
7. trivial vs. non-trivial solutions
8. when does a homogeneous system of linear equations have infinitely many solutions?

Section 1.3: Matrices and Matrix Operations

1. row and column matrices are also called row and column vectors
2. matrices are always capital letters, entries use lower case
3. when are two matrices equal?
4. when (and how) can you add, subtract, or multiply two matrices ? can you divide two matrices?
5. identify the main diagonal of a square matrix
6. partitioned matrices
7. doing matrix multiplication by columns or rows
8. expressing matrix products as a linear combination of rows
9. coefficient matrix of a linear system
10. transpose of a matrix
11. trace of a square matrix

Section 1.4: Inverses

1. observe that most of the familiar arithmetic properties of numbers carry over to matrices
2. but AB ≠ BA in general
3. but AB = AC does not in general allow you to conclude that B = C (when can you “cancel”?)
4. the zero and identity matrices, what are their defining properties?
5. if you have a square matrix, what are the two possibilities for its reduced row-echelon form?
6. invertible vs. singular matrices
7. a matrix can have at most one inverse
8. know the simple formula for the inverse of a 2 × 2 matrix (no shortcuts for 3 × 3 and bigger)
9. if A and B are invertible, what is the inverse of AB?
10. be able to find powers of matrices and know the laws of exponents
11. properties of the transpose, invertibility of the transpose

Section 1.5: Elementary Matrices

1. three kinds of elementary matrices represent three kinds of elementary row operations
2. on what side do you need to multiply elementary matrices?
3. since these row operations are invertible, so are their corresponding elementary matrices
4. what does it mean that two matrices are row equivalent?
5. be able to find the inverse of a matrix using row operations, work column by column from left to
right

Section 1.6: Further Results

1. solving linear systems with the inverse matrix when A is invertible (key idea: no Gaussian elimination)
2. linear systems with a common coefficient matrix but different b’s
3. why do we only need to check one of the two conditions to verify we have an inverse?
4. if AB is invertible with A and B the same size, what can you conclude about A and B?
5. be able to determine consistency by elimination

Section 1.7: Diagonal, Triangular, and Symmetric Matrices

1. be able to find powers and inverses of diagonal matrices
2. identify upper & lower triangular matrices, what are their special properties?
3. what are the properties of symmetric matrices?
4. if A is invertible and symmetric, what can you say about A^-1?
5. if A is invertible, what can you say about AA^T? A^TA?
6. what does it mean to be skew symmetric?

Section 2.1: Determinants

1. know the formulas to calculate the determinant of 2 × 2 and 3 × 3 matrices
2. this idea does not generalize to 4 × 4 matrices and larger!
3. what’s the difference between a cofactor and a minor?
4. we can find the determinant with a cofactor expansion along any row or column
5. what’s the smart way to pick this row/column?
6. how do you find the adjoint from the matrix of cofactors?
7. finding the inverse of A using its adjoint and determinant (key idea: no Gaussian elimination)
8. what’s the easy way to find the determinant of a triangular matrix?
9. solving a system of equations using Cramer’s Rule (key idea: no Gaussian elimination or inverses )

Section 2.2: Determinants by Row Reduction

1. what can you say about the determinant of a square matrix with a row/column of zeros?
2. how are the determinants of A and A^T related?
3. finding the determinant of a triangular matrix is easy
4. how do the elementary row operations affect the determinant of the matrix?
5. what are the determinants of these elementary matrices?
6. what can you say about a square matrix with proportional rows or columns? Why?
7. evaluating determinants by row reduction, a triangular matrix is all you need (but the book prefers
row-echelon)