Review for Linear Algebra Midterm 1
1.1 Introduction to systems of linear equations
• Vocabulary: linear equation, system of equations, solution set/general solution, (in)consistent
system, augmented matrix.
• Determine whether an equation is linear.
• Translate between a system of linear equations and its augmented matrix.
1.2 Gaussian elimination
• Vocabulary: elementary row operations, leading 1, leading variable, free variable, parameter, trivial solution, (reduced) row-echelon form.
• Solve a system of linear equations by applying Gaussian elimination (GE) or Gauss-Jordan
elimination (GJE) to its augmented matrix.
• Determine whether a matrix is in row-echelon form (ref)/reduced row-echelon form (rref)
• Determine whether a system has 0, 1, or infinitely many solutions from the rref of its
augmented matrix.
11.2 Electrical networks
• Vocabulary: electrical circuit, electrical potential, resistance, current, voltage drop.
• Apply Ohm’s Law, Kirchhoff’s Current Law, and Kirchhoff’s Voltage Law to obtain a system of linear equations relating the currents flowing through various parts of an electrical
circuit. Proceed to solve such a system.
1.3 Matrices and matrix operations
• Vocabulary: column matrix, row matrix, scalar, scalar multiplication, main diagonal, linear combination, coefficient matrix, transpose, trace.
• Multiply a matrix by a scalar, add or subtract two matrices.
• If A is an m×n matrix and x an n×1 column matrix, compute Ax as a linear combination
of the columns of A with coefficients the entries of x. If y is a 1 × m row matrix and A an
m × n matrix, compute yA as a linear combination of the rows of A with coefficients the
entries of y.
• Compute the product of an m × n matrix A and an n × p matrix B.
• Translate between a linear system, its augmented matrix [A|b], and the matrix equation
Ax = b.
1.4 Inverses; rules of matrix arithmetic
• Vocabulary: zero matrix, identity matrix.
• Use the properties of matrix arithmetic, and prove them using the definitions of scalar
multiplication, matrix addition and multiplication, and transpose:
[cA]ij
[A + B]ij
[A]ij
= cAij
= Aij + Bij
=
n
X
Aik Bkj
k=1
[AT ]ij
= Aji
• Matrices A and B are inverses if AB = I and BA = I.
• If A and B are invertible matrices of the same size, then (AB)−1 = B −1 A−1 .
• Compute the inverse of an invertible 2 × 2 matrix using the formula
a b
c d
−1
1
d −b
=
.
ad − bc −c a
• If A is invertible, then AT is also invertible with inverse given by
(AT )−1 = (A−1 )T .
1.5 Elementary matrices and a method for finding A−1
• Vocabulary: elementary matrix.
• If an elementary matrix E results from performing a certain elementary row operation
(ERO) on the identity matrix Im , then the product EA is the result of performing the
same ERO on A, for any m × n matrix A.
• If A is an n × n matrix, the following statements are equivalent (TFAE):
1. A is invertible.
2. Ax = b has a unique solution x for any b.
3. rref(A) = In .
4. A can be written as a product of elementary matrices.
1.6 Further results on systems of linear equations and invertibility
• If A is invertible, the equation Ax = b has unique solution x = A−1 b (Theorem 1.6.2).
• If B is a square matrix satisfying either AB = I or BA = I, then the other equation also
holds, hence B = A−1 .
• Let A and B be square matrices of the same size. Then AB is invertible if and only if
both A and B are invertible.
11.6 Markov chains
• Vocabulary: transition probability, transition matrix, state vector, regular transition matrix, steady-state vector.
• Write down the transition matrix for a Markov chain.
• Compute successive state vectors of a Markov chain.
• Determine whether a transition matrix is regular.
• Find the steady-state vector of a regular Markov chain and describe its meaning.
2.1 Determinants by cofactor expansion
• Vocabulary: minor, cofactor, adjoint.
• Compute the determinant of a matrix by cofactor expansion along a row or a column
(Theorem 2.1.1):
n
X
det A =
aij Cij ,
for fixed row i, or
j=1
det A =
n
X
aij Cij ,
for fixed column j.
i=1
• Compute the inverse of a matrix using its adjoint:
A−1 =
1
adj(A).
det(A)
• For a diagonal, upper triangular, or lower triangular matrix, the determinant is the product
of the entries on the main diagonal of the matrix.
• Use Cramer’s Rule to solve a system Ax = b when det A 6= 0.
2.3 Properties of the determinant function
• Use the properties of the determinant. If A, B are n × n matrices, then
det(kA) = k n det(A)
det(A + B) 6= det(A) + det(B)
(in general)
det(AB) = det(A) det(B)
1
det(A−1 ) =
(if A is invertible)
det(A)
det(AT ) = det(A)
• A square matrix A is invertible if and only if det(A) 6= 0.