Properties of Matrix Operations

The main properties of matrix operations — addition, multiplication, transpose, and inverse — are presented below.

In what follows, \( A \), \( B \), and \( C \) are matrices whose sizes are such that the operations are well-defined, \( k \) and \( m \) are scalars, and \( n \) is a positive integer.

Properties of Matrix Addition

  1. \( A + 0 = 0 + A = A \)  (where \( 0 \) is the zero matrix)
  2. \( A + B = B + A \)  (commutativity of matrix addition)
  3. \( (A + B) + C = A + (B + C) \)  (associativity of matrix addition)

Properties of Matrix Multiplication

  1. \( A(BC) = (AB)C \)  (associativity of matrix multiplication)
  2. \( A I_n = A \)  (where \( I_n \) is the identity matrix)
  3. \( I_n A = A \)
  4. \( 0 \cdot A = 0 \)  (where \( 0 \) is the zero matrix)
Important: in general, matrix multiplication is not commutative — \( AB \neq BA \).

Distributive Properties of Matrices

  1. \( A(B \pm C) = AB \pm AC \)
  2. \( (A \pm B) C = AC \pm BC \)

Properties of Matrix Multiplication by Scalars

  1. \( k(A \pm B) = kA \pm kB \)
  2. \( (k \pm m)A = kA \pm mA \)
  3. \( k(mA) = (km)A \)
  4. \( k(AB) = (kA)B = A(kB) \)

Properties of Matrix Transpose

In what follows, \( A^T \) denotes the transpose of \( A \).

  1. \( (A^T)^T = A \)
  2. \( (A \pm B)^T = A^T \pm B^T \)
  3. \( (kA)^T = kA^T \)
  4. \( (AB)^T = B^T A^T \)
  5. \( (I_n)^T = I_n \)
Symmetric matrices. For any real matrix \( A \), the products \( A^T A \) and \( A A^T \) are always symmetric, i.e. \( (A^T A)^T = A^T A \).

Properties of Matrix Inverse

In what follows, \( A^{-1} \) denotes the inverse of \( A \).

  1. \( A A^{-1} = A^{-1} A = I_n \)
  2. \( (A^{-1})^{-1} = A \)
  3. \( (kA)^{-1} = \tfrac{1}{k}A^{-1} \quad (k \neq 0) \)
  4. \( (AB)^{-1} = B^{-1} A^{-1} \)
  5. \( I_n^{-1} = I_n \)
  6. \( (A^T)^{-1} = (A^{-1})^T \)
  7. \( \det(A^{-1}) = \dfrac{1}{\det(A)} \quad (\det(A) \neq 0) \)
Key idea. Matrix multiplication is not commutative, but it is associative, has an identity \( I_n \), and satisfies the two reversal rules: \[ (AB)^T = B^T A^T, \qquad (AB)^{-1} = B^{-1} A^{-1}. \]

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