Course Intro

Math 303: Section 16

Dr. Janssen

About this course

About Dr. Janssen

• Tenth year at Dordt
• Ph.D. in math from Nebraska
• Family: Laura, Lila (7), Sam (4), Arthur (🐈)
• Interests: running

Summer 2023 at Custer State Park

Course Overview

• A continuation of Math 203
• A deeper exploration of vector spaces, matrices, and orthogonality
• Richer applications
• Plan: Cover roughly Sections 16–34¹

Main challenge/opportunity: different backgrounds with this material, at different times. As we integrate last year’s linear algebra class with this year’s, I’ll ask for your patience, both if something seems too tedious, and if it seems things are moving too fast. Please let me know if, and speak up if you have questions, concerns, etc.

A silly issue: version of the text from fall 2023 is slightly reorganized from the version used in fall 2022.

Some unique opportunities!

1. We’ll start fast, and slow down as we go.

Course Liturgies

• Preview Activities
• Lecture
• In-class Activities
• Edfinity Homework (access code required)
• Labs
• Quizzes
• Exams/Project

Questions?

Things to remember

Notation

• By \(\mathbb{R}\) we mean the set of real numbers; \(\mathbb{R}^n\) denotes the set of \(n\)-component vectors with real entries.
• In general, upper case letters like \(A, B, C, D, P\) denote \(m\times n\) matrices
• \(m\): number of horizontal rows
• \(n\): number of vertical columns
• \(a_ij\) in the matrix \(A\) denotes the _entry_in row \(i\), column \(j\).
• Vectors are denoted by \(\mathbf{v}\), \(\mathbf{w}\), etc, and are generally written as columns: \(\mathbf{v} = \left[\begin{matrix} -1 \\ 4 \end{matrix}\right]\).
• So much depends on solving equations like \(A\mathbf{x} = \mathbf{b}\).

Like what?

\(\S\) 16: The Determinant

Example

Example 1 (The \(2\times 2\) case) Let’s calculate

\[ \det \left[\begin{matrix} 1 & 4 \\ 2 & -2 \end{matrix}\right]. \]

Sage: https://sagecell.sagemath.org

Code

A = Matrix([[1,4],[2,-2]])
A, A.det()

(
[ 1  4]     
[ 2 -2], -10
)

In general, it’s defined inductively.

A more interesting example

Example 2 (A \(3\times 3\) example) Let’s calculate

\[ \det \left[\begin{matrix} 1 & 0 & -3 \\ 1 & 2 & 4 \\ 2 & -1 & -2 \end{matrix}\right] \]

Code

B = Matrix([[1,0,-3],[1,2,4],[2,-1,-2]])
B, B.det()

(
[ 1  0 -3]    
[ 1  2  4]    
[ 2 -1 -2], 15
)

The formal definition

Definition 1 (Cofactor definition of the determinant) Let \(A\) be an \(n\times n\) matrix, and denote by \(A_{ij}\) the \((n-1)\times (n-1)\) matrix obtained by deleting row \(i\), column \(j\) of \(A\). The \(ij\)th cofactor of \(A\), denoted \(C_ij\), is the number \(C_{ij} = (-1)^{i+j} \det(A_{ij})\).

The determinant of \(A\) is then the number

\[ \det(A) = a_{11} C_{11} + a_{12} C_{12} + \cdots + a_{1n} C_{1n}. \]

Note that the definition of the determinant involves taking the cofactor expansion along the first row. It is a fact that we can actually expand along any row or column and get the same thing. This is most often useful when there is a row/column that contains a lot of zeros.

Explore!

Exercise 1 (All about determinants)  

1. Let \(A = \left[\begin{matrix} 1 & 3 & 1 \\ 0 & 0 & 2 \\ -2 & 4 & 5 \end{matrix}\right]\). Use cofactor expansion along the first row to calculate \(\det A\).

2. Calculate the determinant of \(A\) along the second row and confirm that you get the same thing as you did along the first.

3. Calculate the determinant of \(B = \left[\begin{matrix} 0 & 0 & 2 \\ 1 & 3 & 1 \\ -2 & 4 & 5 \end{matrix}\right]\) using cofactor expansion along the first row. Compare your answer to what you found in the previous two parts.

4. Calculate the determinant of \(C = \left[\begin{matrix} 1 & 3 & 1 \\ 0 & -2 & 3 \\ 0 & 0 & 5 \end{matrix}\right]\). What do you notice/wonder/remember?

A = Matrix([[1,3,1],[0,0,2],[-2,4,5]])
B = Matrix([[0,0,2],[1,3,1],[-2,4,5]])
C = Matrix([[1,3,1],[0,-2,3],[0,0,5]])
A.det(), B.det(), C.det()

(-20, 20, -10)

Properties of the determinant

Theorem 1 (Properties of the Determinant) Given \(n\times n\) matrices \(A,B\), the following hold:

• \(\det(AB) = \det(A) \det(B)\)
• \(\det(A^\textsf{T}) = \det(A)\)
• \(A\) is invertible if and only if \(\det(A)\ne 0\)
• If \(A\) is invertible, then \(\det(A^{-1}) = 1/\det(A)\)
• If \(A = \left[\begin{matrix} a & b \\ c & d \end{matrix}\right]\), then \(\det(A) = ad - bc\).
• If \(A\) is upper/lower triangular, then \(\det(A)\) is the product of the diagonal entries.
• The determinant of a matrix is the product of the eigenvalues, which each eigenvalue repeated according to its multiplicity.
• Adding a multiple of a row to another row does not change the determinant.
• Multiplying a row by a constant multiplies the determinant by the same constant.
• Row swapping multiplies the determinant by \(-1\).

Theorem 16.2 in the text.

More on eigenvalues to come.

As the size of the matrices increases, the number of \(2\times 2\) determinants required for cofactor expansion increases in a hurry. So it’s often preferable to row reduce, keeping track of the operations and their effects on the determinants.

Our main use of the determinants (early in the semester, at least), will be for finding eigenvalues by hand, and as a theoretical tool for working with (non)invertible matrices.