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MAT223 Lecture 07

MAT223 Lecture 07 Raw
MAT223 Lecture 07 Flashcards

• Completed Notes Status

  • Completed insertions: 2
  • Ambiguities left unresolved: 1 (Arithmetic discrepancy in determinant example)

• Lecture Summary

  • Central objective: Extend matrix algebra to bilinear forms, introduce the determinant for $n\times n$ matrices via cofactor expansion, and explore properties of symmetric and skew-symmetric matrices.
  • Key concepts:
    • Bilinear Forms: Expressing sums like $\sum \sum a_{ij}{x}_j{y}_i$ as matrix products $(A\overrightarrow{x})\cdot \overrightarrow{y}$.
    • Determinants: Calculating determinants using Cofactor Expansion (Laplace expansion), specifically utilizing rows/columns with zeros to simplify calculation.
    • Symmetry: Defining Symmetric Matrix ($A^T=A$) and Skew-Symmetric Matrix ($A^T=-A$).
  • Connections:
    • The definition of Multiplying Matrices is strictly tied to dimension compatibility (inner dimensions must match).
    • Constructing symmetric matrices from arbitrary ones using $A+A^T$ and skew-symmetric ones using $A-A^T$.

• TK Resolutions

  • #tk: Try creating a question where we have some information about a matrix $A$ and a cofactor $A_{ij}$, and we have to re-derive $A$.
    • Answer: This is generally impossible because the cofactor $A_{ij}$ (strictly the determinant of the submatrix) or the submatrix itself loses the information contained in the $i$-th row and $j$-th column.
    • Revised Question for Study: "Given the submatrix $A_{11}$ of a $3\times 3$ matrix $A$, and knowing $A$ is symmetric with diagonal entries equal to 1, can you reconstruct $A$?" (This adds constraints that might make it solvable).

• Practice Questions

  • Remember/Understand:
    • What is the condition on matrix dimensions $n,m,k,l$ for both $AB$ and $BA$ to be defined?
    • Define a Skew-Symmetric Matrix in terms of its transpose.
  • Apply/Analyze:
    • Prove that for any square matrix $A$, the matrix $B=A-A^T$ is skew-symmetric.
    • Compute the determinant of $A=\left[\begin{array}{ccc}2& 0& 3\\ 1& 5& 1\\ 0& 0& 2\end{array}\right]$ using cofactor expansion along the most efficient row.
  • Evaluate/Create:
    • Why is the dot product not defined for $2\times 2$ matrices in the standard sense used in this course?
    • Construct a $3\times 3$ matrix that is both symmetric and skew-symmetric. (Hint: Check the zero matrix).

• Challenging Concepts

  • Cofactor Expansion:
    • Why it's challenging: Keeping track of indices and the alternating signs $(-1{)}^{i+j}$ during recursive calculations.
    • Study strategy: Practice drawing the "checkerboard" of signs ($+-+\dots$) over the matrix before starting the expansion. Always choose the row/column with the most zeros.
  • Skew-Symmetric Matrix:
    • Why it's challenging: The diagonal elements must be zero ($a_{ii}=-a_{ii}⟹2a_{ii}=0$), which is often forgotten.
    • Study strategy: Check the diagonal first. If any diagonal element is non-zero, it cannot be skew-symmetric.

• Action Plan

  • Immediate review actions:
    • Review all [COMPLETED] insertions and verify with course materials
    • Address all #tk items (if any)
    • 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 only where the new lecture adds content
  • 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