Matrices and Determinants

Rank, Echelon Form and Applications

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Rank, Echelon Form and Applications

Rank, Echelon Form and Applications

This lesson covers the following key concepts:

  • Echelon and reduced echelon form
  • Rank of a matrix
  • Elementary row/column operations
  • Properties of rank
  • Relationship: rank and system of equations
  • Geometry applications: area, collinearity, straight line
  • Advanced JEE applications

Important Formulas

  • Rank(A)=number of non-zero rows in echelon form\text{Rank}(A) = \text{number of non-zero rows in echelon form}
  • Elementary operations preserve rank\text{Elementary operations preserve rank}
  • Rank(AB)≤min⁡(Rank(A),Rank(B))\text{Rank}(AB) \leq \min(\text{Rank}(A), \text{Rank}(B))
  • Area of triangle=12∣x1y11x2y21x3y31∣\text{Area of triangle} = \frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}
  • Collinear iff det = 0\text{Collinear iff det = 0}
  • Line equation: ∣xy1x1y11x2y21∣=0\text{Line equation: } \begin{vmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{vmatrix} = 0