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1. Matrix Operations

Transpose (AT): Interchanging rows and columns. If A =

a   b
c   d
, then AT =
a   c
b   d
.

Multiplication (AB): Multiply row of A by column of B. Ensure ColsA = RowsB. Note: AB ≠ BA.

2. Determinant & Inverse

Determinant |A| = ad - bc
Common Pitfalls:
• **Singular Matrices:** If |A| = 0, stop! No inverse exists.
• **Adjoint Signs:** Don't forget to negate the secondary diagonal (b and c) when finding the Adjoint matrix.

Inverse Formula: A-1 = (1 / |A|) *

d   -b
-c   a

3. Linear System Solvers

For the system: a1x + b1y = c1 and a2x + b2y = c2

A. Cramer's Rule (Determinant Method)

A direct approach using determinant ratios. Avoids the inverse matrix entirely.

  • D:
    a₁   b₁
    a₂   b₂
    = a₁b₂ - a₂b₁
  • Dx:
    c₁   b₁
    c₂   b₂
    = c₁b₂ - c₂b₁
  • Dy:
    a₁   c₁
    a₂   c₂
    = a₁c₂ - a₂c₁
x = Dx / D   ,   y = Dy / D

B. Matrix Inverse Method

Transforms the system into the equation AX = B.

  1. Setup:
    a₁   b₁
    a₂   b₂
    x
    y
    =
    c₁
    c₂
  2. Isolate X: Multiply by A-1 from the left: X = A-1B.
X = (1 / |A|) * Adj(A) * B