CHAPTER 3 & 4 Study Notes on Matrices and Vector Spaces

CHAPTER 3. MATRICES

3.11. DEFINING DETERMINANTS

We can compute $ext{det}(A)$ :
ext{det}(A) = extstyleiggl(egin{array}{c} p \ S_2 \ ext{det} A \ ext{det}(A) \= extstyle{iggl(iggl)iggl)
By the definition of determinant:
$ext{det}(A) = extstyle\biggl( extstyle\biggl) ext{det}(A)$
   ext{det}(A) = extstyleiggl(egin{array}{c} p \ S_2 \ iggig{(} ext{det}(Aigg)igg)igg) + (-1)^ ext{p_{i,j}} ext{det}(p_{i,j})
  - Exercise 3.11.24 : Use the definition of the determinant to calculate
  igg| egin{array}{cccc} 0 & 1 & 1 + ext{p}{2} & 2 + ext{p}{2} \ 2 + ext{p}_{2} & 2 + 3 ext{p} \ -1 & ext{p} igg|
  - Remark: It is noted that $| ext{det}(A)|$ only applies to $n imes n$ determinants with n > 1!
  - Exercise 3.11.25: Show that $ext{det}[-6] <br />\neq | - 6|$ .

3.12. DETERMINANT PROPERTIES

3.12 Determinant Properties

Theorems on shortcuts for computation of determinants:(replace with C for less general case)
Theorem 3.12.1 (Diagonal/Triangular Determinant Theorem)
- If A = egin{pmatrix} a_{ij} ext{ for } i, j = 1 \ ext{if a_{d}} ext{ for any n } \ ( ext{det}(A)) = extstyleiggl( ext{det}(A)
Example 3.12.1: $I_2$ is both lower and upper triangular, thus $ext{det}(I_2) = (1)(1) = 1$ .
Exercises:
- Exercise 3.12.1: Use the Diagonal/Triangular Determinant Theorem to compute: egin{pmatrix} 1 & -5 \ 0 & 4 \ ext{det}(A) \ = …
Theorem 3.12.2 (Elementary Row Operation Determinant Theorem):
  - Part (a): By switching two rows: $ext{det}(B) = - ext{det}(A)$ .
  - Part (b): Multiplying one row of $A$ by any constant $z$ : $ext{det}(B) = z ext{det}(A)$ .
  - Part (c): Adding a multiple of one row to another row does not affect the determinant: $ext{det}(B) = ext{det}(A)$ .
Example 3.12.2: Determine egin{pmatrix} 0 & 1 \ 1 & 0 \ ext{det}(B) \ …
- Result: Since it is achieved by switching two rows, the result is negative, implying that $ext{det}(I_2) = - ext{det}(I) = -1$ .
Theorem 3.12.3 (Determinant-Invertible Matrix Theorem)
- If $A ext{ in } M_n$ , $det(A) <br />\neq 0 ext{ t})$ .
Example:
- egin{pmatrix} ext{det}(A) = >>>> = int(-5) >>…
- By the theorem, if any matrix contains a zero row, its determinant must be zero.
Theorem 3.12.4 (Homogeneous System-Determinant Solution Quantification Theorem)
- (a): The unique solution $0_n$ holds only if $det(A) <br />\neq 0$ , thus yielding infinite solutions.

- (b): $Ax = 0_n$ has unique solution if truth holds.

3.13. COFACTORS AND MINORS

3.13 Cofactors and Minors

Definition: For any matrices $A ext{ in } M_n$ , (i,j)-minor $M_{i,j}(A)$ defined as $ext{det}(A) = ext{det}(A^{(n-1)})$ and assigns a sign $(-1)^{i+j}$ for cofactor $C_{i,j}(A)$ by $C_{i,j}(A) = (-1)^{i+j}M_{i,j}(A)$.
Example: Consider a matrix $A ext{ in } M_n$ :
- For $I ext{ in } M_2$ … with values $1, 2$ determining the minors.
- Verify values of cofactors of cofactor matrix.
Theorem: By definition, $|det A|$ … follows from those determinants being nonzero.
Therefore, obtaining useful properties provides necessary aspects for working determinants with linear algebra directly.

Additional Details

In the next chapters, further examination into spans, linear dependency, independence, basis, and within cases concerning homogeneous linear systems and general multidimensional spaces will allow insights into solving differential equations and applying polynomial characteristics to vector algebra cases for advanced mathematics…

This concludes Chapter 3 and starts further explorations into matrix application.