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MAT223 Lecture 09

MAT223 Lecture 09 Raw
MAT223 Lecture 09 Flashcards

• Completed Notes Status

  • Completed insertions: 3
  • Ambiguities left unresolved: none

• Lecture Summary

  • Central objective: Apply Determinant properties to solve equations involving matrix products, transposes, and inverses; introduce Eigenvalues, Eigenvectors, the Characteristic Polynomial, and diagonalizability
  • Key concepts:
    • Determinant properties: $det(A^t)=det(A)$^[1], $det(A^{-1})=\frac{1}{det(A)}$^[1:1], $det(cA)=c^{n}det(A)$ for an $n\times n$ matrix, and $det(AB)=det(A)det(B)$^[1:2]
    • If both determinants $det(A)=0$ and $det(B)=0$, then $det(AB)=0$ (the product of singular matrices is singular)
    • Eigenvalue and Eigenvector: A scalar $\lambda$ and non-zero vector $\overrightarrow{v}$ satisfying $A\overrightarrow{v}=\lambda \overrightarrow{v}$
    • Characteristic Polynomial $c_A(x)=det(x{I}_n-A)$: the roots of this polynomial are the Eigenvalues of $A$^[2][3]
    • Diagonalizable Matrix: An $n\times n$ matrix $A$ is diagonalizable if and only if $A$ has $n$ linearly independent Basic Eigenvectors^[4][5]
  • Connections:
    • Determinant properties underpin solving matrix equations and establish the relationship between Eigenvalues and the Characteristic Polynomial
    • The existence of Eigenvalues guarantees at least one Basic Eigenvector per eigenvalue, and the count of linearly independent Basic Eigenvectors determines diagonalizability

• TK Resolutions

  • No #tk flags present in the raw input

• Practice Questions

  • Remember/Understand:
    • What is the relationship between $det(A)$ and $det(A^t)$?
      • $detA=det{A}^t$
    • If $\lambda$ is an Eigenvalue of matrix $A$, what equation must $\lambda$ satisfy?
    • Define the Characteristic Polynomial of an $n\times n$ matrix $A$
  • Apply/Analyze:
    • Given $det(A)=2$ and $det(B)=-3$ for $2\times 2$ matrices, compute $det(2A^2(B^t{)}^{-1})$
    • Determine whether the matrix $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ is diagonalizable by finding its Characteristic Polynomial and Eigenvalues
    • Prove that if $det(A)=0$ and $det(B)=0$, then $det(AB)=0$
  • Evaluate/Create:
    • Construct a $2\times 2$ matrix with Eigenvalues $\lambda =5$ and $\lambda =-2$, and verify your answer by computing the Characteristic Polynomial
    • Explain why a matrix with fewer than $n$ linearly independent Basic Eigenvectors cannot be diagonalizable

• Challenging Concepts

  • Determinant of scalar multiples:
    • Why it's challenging: Confusion between $det(cA)$ and $c\cdot det(A)$; the scalar $c$ affects each row, yielding $c^{n}det(A)$ for an $n\times n$ matrix
    • Study strategy: Practice computing $det(c{I}_n)$ for various $n$ to internalize the $c^n$ factor; work through problems like Question 4 in the raw notes
  • Diagonalizability criterion:
    • Why it's challenging: Recognizing that an $n\times n$ matrix requires exactly $n$ linearly independent Basic Eigenvectors to be diagonalizable, which may not always occur even when $n$ Eigenvalues exist (counting multiplicity)
    • Study strategy: Compare cases where a matrix has $n$ distinct Eigenvalues (always diagonalizable) versus repeated Eigenvalues (may or may not be diagonalizable)

• Action Plan

  • Immediate review actions:
    • Review all [COMPLETED] insertions and verify with course materials
    • Re-read Lecture Summary and confirm key links
  • Practice and application:
    • Answer Remember/Understand questions
    • Answer Apply/Analyze questions
    • Attempt Evaluate/Create questions
  • Deep dive study:
    • Focus on Challenging Concepts (above)
    • Extend atomic notes for Determinant, Eigenvalue, Characteristic Polynomial, and Diagonalizable Matrix
  • Verification and integration:
    • Cross-reference notes with textbook/slides
    • Identify remaining gaps
    • Link this lecture to prior/future lecture notes (lecture notes only; no MOCs)

• Footnotes

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