AHL 1.14 MATRICES

AHL 1.14: MATRICES

๐Ÿ“Œ Key Definitions

Term Definition Example
Matrix A rectangular array of numbers arranged in rows and columns. Example: [1 2]
[3 4]
Element The number at the intersection of a row and column in a matrix (notation: Aij). 2 is element (1,2) of [1 2]
[3 4]
Order The matrix’s size: number of rows ร— number of columns (m ร— n). Above matrix is order 2 ร— 2
Identity Matrix Square matrix with 1s on diagonal and 0s elsewhere; acts as the multiplicative identity. I = [1 0]
[0 1]
Zero Matrix Matrix with all entries equal to zero; additive identity. 0 = [0 0]
[0 0]

๐Ÿ”ข GDC & Technology Integration

Use the calculator’s matrix editor to input matrices (confirm rows ร— columns). Use built-in determinant, inverse and linear-solve functions. Store matrices in memory to avoid re-entry; verify inverses by computing A ร— Aโˆ’1 = I. For larger problems, spreadsheets or CAS systems allow symbolic manipulation and numerical stability checks.

๐Ÿ“Œ Algebra of Matrices

  • Equality: A = B if orders match and Aij = Bij for all i,j.
  • Addition: (A + B)ij = Aij + Bij (defined only for same order).
  • Subtraction: (A โˆ’ B)ij = Aij โˆ’ Bij.
  • Scalar multiplication: (kA)ij = k ร— Aij.

๐Ÿ” TOK Perspective

Matrices compress relational information into compact structures. Discuss whether the abstraction hides assumptions (e.g. linearity) and how that affects trust in model outputs. Does mathematical compactness enhance or reduce explanatory power?

๐Ÿ“Œ Multiplication of Matrices

  • To multiply A (m ร— n) by B (n ร— p): (AB)ij = ฮฃk=1..n Aik ร— Bkj.
  • Multiplication is associative: (AB)C = A(BC), and distributive over addition, but generally not commutative (AB โ‰  BA in general).
Example: A = [1 2]
[3 4], B = [2 0]
[1 3] โ†’ AB computed entrywise:

AB = [ (1ร—2 + 2ร—1) , (1ร—0 + 2ร—3) ]
[ (3ร—2 + 4ร—1) , (3ร—0 + 4ร—3) ] = [4 , 6] [10 , 12].

๐ŸŒ EE Focus

An EE might investigate the stability of repeated matrix multiplication (e.g., Markov chains) or the effect of rounding error in iterative matrix methods โ€” both rich topics linking linear algebra theory to numerical analysis.

๐Ÿ“Œ Identity & Zero Matrices

  • Identity I: A ร— I = I ร— A = A for conformable sizes.
  • Zero 0: A + 0 = 0 + A = A.

๐Ÿ“Œ Determinants & Inverses

  • 2 ร— 2 determinant: |A| = a d โˆ’ b c for A = [a b]
    [c d].
  • Inverse (if |A| โ‰  0): Aโˆ’1 = (1 รท |A|) ร— [d โˆ’b]
    [โˆ’c a].
Example: A = [4 7]
[2 6] โ†’ |A| = 24 โˆ’ 14 = 10 โ†’ Aโˆ’1 = (1/10)[6 โˆ’7]
[โˆ’2 4].

๐Ÿงฎ IA Tips & Guidance

Use real datasets to model systems (economic input-output, population transitions) and demonstrate how singular matrices (|A|=0) reveal dependencies or lack of invertibility. Discuss limitations and numerical stability when computing inverses with technology.

๐Ÿ“Œ Solving Linear Systems

  • Matrix form: A x = b, where A is coefficient matrix, x vector of unknowns, b constants.
  • If Aโˆ’1 exists: x = Aโˆ’1 b. If A singular, investigate consistency (no solution or infinitely many).

๐Ÿง  Examiner Tip

Practice both by-hand methods (2ร—2 inverses, determinant checks) and GDC workflows (matrix entry, inverse, solve). In exam answers, indicate the order and clearly state intermediate checks (e.g., Aร—Aโˆ’1=I).