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

Math 303: Section 16

Dr. Janssen

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

- A continuation of Math 203
- A deeper exploration of vector spaces, matrices, and orthogonality
- Richer applications
- Plan: Cover roughly Sections 16–34
^{1}

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

Questions?

- 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 \(A\mathbf{x} = \mathbf{b}\).

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

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

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

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

**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

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

**Exercise 1 (All about determinants) **

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

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.

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?

**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 \(-1\).