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Cheat sheet · JEE Preparation Mathematics

Matrices cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

JEE · Mathematics

Cheat sheet

Class 12 · Chapter 3Matrices

(AB)′ = B′A′ — the order reverses.

1. Types

  • Row, column, square, diagonal, scalar, identity, zero.
  • Symmetric: A′=AA' = A; skew-symmetric: A′=−AA' = -A (diagonal zero).
  • Every square matrix = 12(A+A′)+12(A−A′)\tfrac12(A + A') + \tfrac12(A - A').

2. Operations

  • Addition needs same order.
  • AB defined if columns of A = rows of B.
  • AB ≠ BA in general.
  • AB = O doesn't mean A = O or B = O.

3. Transpose & Inverse

(A′)′=A(A')' = A(A+B)′=A′+B′(A + B)' = A' + B'(AB)′=B′A′(AB)' = B'A'
AA−1=A−1A=IAA^{-1} = A^{-1}A = I(AB)−1=B−1A−1(AB)^{-1} = B^{-1}A^{-1}

Inverse by elementary row (or column) operations: [A | I] → [I | A⁻¹].

4. Key Points

  • Order m × n has mn elements.
  • Identity is a scalar matrix.
  • Inverse is unique if it exists.
  • (kA)′ = kA′.

Remember: Types | multiplication rules | (AB)′ = B′A′

Matrices: questions and notes

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