Properties of matrix multiplication

Lessons

• Introduction
  Properties of matrix to matrix multiplication overview:
  a)

  The basic matrix to matrix multiplication properties
  
  
  b)

  Failure of the Commutative property
  
  
  c)

  Dimension property
  
  

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• 1.
  Verifying the properties of matrix to matrix multiplication
  You are given that . Verify that:
  a)

  $(XY)Z=X(YZ)$
  
  
  b)

  $X(Y+Z)=XY+XZ$
  
  
  c)

  $(Y+Z)X=YX+ZX$
  
  
  d)

  $OX=O$
  
  
  e)

  $IY=Y$
  
  

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• 2.
  Showing that the Commutative property fails
  You are given that . Show that:
  a)

  $XY \neq YX$
  
  
  b)

  $X(Y+Z) \neq (Y+Z)X$
  
  

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• 3.
  Dimension Property
  You are given that $X$ is a 2 x 4 matrix, $Y$ is a 3 x 3 matrix, and $Z$ is a 4 x 3 matrix. Are the following defined? If it is defined, show the dimensions of the matrix.
  a)

  $XY$
  
  
  b)

  $XZ$
  
  
  c)

  $ZX$
  
  
  d)

  $ZY$
  
  
  e)

  $XZY$
  
  
  f)

  $ZYX$
  
  

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