Properties of Determinants
Goals: To prove Det(A) = 0 iff A is singular.
Det(AB) = Det(A)Det(B)
To calculate Det(A) by factoring A = L U
After the determinant is defined, the exploration and discovery process begins. Mathematicians attempt to formalize the results of their conjectures as a set of theorems. We summarize the results that we will need later in the course.
A Few Theorems
This part of the class is devoted to giving you practice reading and writing proofs. You will be asked to turn in a proof of one of our conjectures. We will freely use the cofactor expansion for evaluating determinants.
Prove: If In is the nxn identity matrix, the Det(In) = 1.
Pf. Exercise: Use mathematical induction on the cofactor expansion.
Rather than using Leibniz's Formula for our proofs about determinants we will use Laplace's method of cofactor expansion. In order to do this we need the following lemma.
Lemma
Let A be an nxn matrix. If
denotes the cofactor of
for k = 1, ... , n, then
if i = j
= {
Eq. 1
if
Before we prove this result, we illustrate the statement with an example.
Let
i =1 and j = 2. The Eq. 1 is written out as:
=
This is by the definition of cofactors.
Expanding we get,
=
after some algebra,
The Proof of Lemma
if i = j, Eq. 1 is simply the cofactor expansion of the Det(A).
if
then Eq. 1 is equivalent to replacing the jth row with a copy of the ith row and expanding about the jth row. Since the determinant of a matrix with two equal rows is 0, we have proven our lemma.
Show that Eq. 1 is equivalent to replacing the jth row with a copy of the ith row and expanding about the jth row.
the jth row
Suppose
the ith row
Notice the first term of the expansion is:
Det(E A)
What is the effect on the determinant of a matrix if we multiply it first by an elementary matrix? This is what is Det(EA)? We first consider a type 2 elementary matrix, that is a row of A is multiplied by a non zero constant.
The Pattern
Define
Theorem: If E2 is an elementary matrix that corresponds to multiplying on row of a matrix by a nonzero constant a, then Det(E2A) = a Det(A) = Det(E2)Det(A) and Det(E2) = a.
Proof: Expand Det(E2A) about the row that is multiplied by a.
=
=
Since
We can write:
q.e.d.
The Pattern
Define
Theorem: If E3 is an elementary matrix that corresponds to the type three elementary row operation that takes a multiple of a row and adds it to another row, then
and
Proof: Let the jth row of the matrix A be the row affected by the E3 row operation. Expand
about the jth row, that is
=
By our Lemma,
=
Now since,
we have
Thus,
and
q.e.d.
The Pattern
Define
Theorem: If E1 corresponds to a type I row operation, that is switching two rows of a matrix,
then
and
Proof: Exercise. The proof of this theorem can be accomplished by writing the E1 matrix as a series of E2 and E3 matrix multiplications, then by using the results from the previous two theorems we can compute
.
Theorem: If E is an elementary matrix, then
Proof:
q.e.d.
Theorem: If
is a product of elementary matrices then
Proof:
We are ready now to prove the first statement of our goal.
Theorem: An nxn matrix A is singular if and only if Det(A) = 0.
Proof: The matrix A can be reduced to row echelon form U with a finite number of elementary row operations. Thus
where U is in row echelon form and each of the
's are elementary matrices.
If A is singular then U contains a row of zeroes. Thus Det(U) = 0. But,
Since
for any of the elementary matrices, the Det(A) = 0.
q.e.d.
Finally, we prove our last theorem.
Theorem: If A and B are matrices, then Det(AB) = Det(A)Det(B).
Proof: If B is singular, it follows from a theorem that AB is also singular and therefore
If A and B are nonsingular, then B can be written as a product of elementary matrices, and
=
=
q.e.d.
Aren't the theorems an elegant summation? If you made some conjectures that are not proven here, try to write a proof of your result.
Some Applications
The Cross Product of two vectors in R3.
The Area of a Quadrilateral
____________________________________________________
Lecture 5 Notes
