Properties of Matrix Operations

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The main properties of matrix operations such as addition, multiplication, transpose and inverse are presented.
In what follows, \( A, B \) and \( C \) are matrices whose sizes are such that the operations are well defined and \( k \) and \( m \) are scalars.
\( I_n \) is the identity matrix of size \( n \times n \) whose diagonal entries are all equal to 1 and all non diagonal entries equal to zero.
\( 0 \) is the zero matrix whose entries are all zeros.


Properties of Matrix Addition

  1.    \( A + 0 = 0 + A = A\) , where \( 0 \) is the zero matrix.
  2.    \( A + B = B + A \)
  3.    \( (A + B) + C = A + (B + C) \)


Properties of Matrix Multiplication

  1.    \( (A B) C = B(A C) \)
  2.    \( A I_n = A \) , where \( I_n \) is the identity matrix.
  3.    \( I_n A = A \)
  4.    \( 0 A = 0 \) , where \( 0 \) is the zero matrix.
  5.    In general \( AB \ne BA \)


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


Properties of Matrix Transpose

  1.    \( (A^T)^T = A \)
  2.    \( (A \pm B)^T = A^T \pm B^T \)
  3.    \( ( k A)^T = k A^T \)
  4.    \( (A B)^T = B^T A^T \)
  5.    \( (I_n)^T = I_n \)


Properties of Matrix Inverse

  1.    \( A A^{-1} = A A^{-1} = I_n \)
  2.    \( (A^{-1})^{-1} = A \)
  3.    \( (k A)^{-1} = k^{-1} A^{-1} = \dfrac{1}{k} A^{-1} \) , for \( k \ne 0 \)
  4.    \( (A B)^{-1} = B^{-1} A^{-1} \)
  5.    \( I_n^{-1} = I_n \)
  6.    \( (A^T)^{-1} = (A^{-1})^T \)



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