Matrix Algebra

Matrix Algebra

We review here some of the basic definitions and elementary 

algebraic operations on matrices. There are many applications 

as well as much interesting theory revolving around these concepts, 

which we encourage you to explore after reviewing 

this tutorial.

A matrix is simply a retangular array of numbers. For example,

A=      a11 a21  am1 a12 a22  am2     a1n a2n  amn        

is a mn matrix (m rows, n columns), where the entry in the ith row and jth column is aij. We often writeA=[aij].

Key Concepts

Let A=[aij] and B=[bij].

• Transpose AT of A:
  
  AT=[aji]
  
  
• Trace of A:
  
  nk=1akk (for an nn matrix A) 
  
  
• Identity Matrix I:the nn matrix with 1's on the main digonal and 0's elsewhere.
• A+B and A−B:
  
  A+B=[aij+bij]
  A−B=[aij−bij]
  
  
• Scalar Multiplication:
  
  cA=[caij]
  
  
• Matrix Product AB:(ij)th entry is nk=1aikbkj 
  (for an mn matrix A and an np matrix B).
• Inverse A−1 of A:A−1 satisfies AA−1=A−1A=I.
  If A=a c b d   ,
  then A−1=1ad−bcd −c −b a   
• Determinant detA:If A=a c b d   , detA=ad−bc.
  In general,
   along row i:
   detA=ai1ci1(A)+ai2ci2(A)++aincin(A).
   along column j:
   detA=a1jc1j(A)+a2jC2j(A)++anjcnj(A).