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03Class 12 Maths

Matrices formulas

Matrix types and operation rules. Matrix multiplication behaves differently from ordinary numbers, and that's where most errors start.

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All Maths formulas

Operations

Product condition
Am×nBn×p=Cm×p\displaystyle A_{m \times n}B_{n \times p} = C_{m \times p}
Not commutative
AB≠BA in general\displaystyle AB \ne BA \ \text{in general}
Transpose rules
(A′)′=A,(A+B)′=A′+B′,(AB)′=B′A′\displaystyle (A')' = A, \quad (A + B)' = A' + B', \quad (AB)' = B'A'

Special matrices

Symmetric
A′=A\displaystyle A' = A
Skew-symmetric
A′=−A\displaystyle A' = -A

Diagonal entries are all zero.

Any square matrix
A=12(A+A′)+12(A−A′)\displaystyle A = \tfrac12(A + A') + \tfrac12(A - A')

Sum of a symmetric and a skew-symmetric matrix.

Inverse
AA−1=A−1A=I\displaystyle AA^{-1} = A^{-1}A = I
Inverse of a product
(AB)−1=B−1A−1\displaystyle (AB)^{-1} = B^{-1}A^{-1}

Where students slip up

  • Cancelling matrices. AB = AC doesn't mean B = C unless A is invertible.
  • Writing (AB)' as A'B'. The order reverses.
  • Multiplying matrices whose orders don't match.

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