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matrix operations
1 Algebraic properties of matrix operations
1.1 Properties of matrix addition
Let A, B, C be m× n matrices  Let a ,b, c be any numbers  
Associative  
Commutative  
3. A+ O = A, where O = [0] is the zero matrix . 
a+0 = a.  Additive identity 
4. A+(−A) = O, where −A = [−a_{ij} ] is the negative of A. 
a+(−a) = 0.  Additive inverse 
1.2 Properties of matrix multiplication
Let A, B, C be appropriate matrices  Let a ,b, c be any numbers  
1. A(BC) = (AB)C  a(bc) = (ab)c  Associative 
2. A(B+ C) = AB+ AC C(A+ B) =CA+ CB 
a(b+ c)=ab+ ac =(b+ c)a  Distributive 
AB may not be BA  ab = ba.  Commutative 
4. AI = IA = A, only for square matrices, I is called the idetity. 
a 1 = a  Mult. identity 
5. A^{−1} exits only when they are invertible 
Mult. inverse 
Example. Example 10 page 39, pblm 32 page 41.
1.3 Properties of scalar multiplication
Let r, s ∈R and A ,B be appropriate matrices
Associative and Commutative  
Distributive 
1.4 Properties of transpose
Let r ∈ R and A, B be appropriate matrices
Self inverse  
Linear  
2 Special types of Matrices
2.1 Square matrices
Definition. Let A = [a_{ij} ] be a square matrix of order n (n× n
type), we say
1. A is a upper triangular matrix if a_{ij} = 0 for i > j.
2. A is a lower triangular matrix if a_{ij} = 0 for i < j.
3. A is a symmetric if A^{T} = A and skew symmetric if A^{T }
= −A.
4. A is a diagonal matrix if a_{ij} = 0 for i ≠ j.
5. A is a scalar matrix if A is diagonal and all elements on the diagonal
are equal to each others, i.e., a_{ij} = c for some c ∈R
and for all i = 1,2, . . . ,n.
6. A is an identity matrix if A is scalar and the scalar is 1. We denote
I_{n} the identity matrix of order n.
Powers of square matrix
Definition. Let A be a square matrix of order n. Let p be an nonnegative
integer, we define the pth power of A, A^{p} by
Properties.
2.2 Nonsingular matrices
Definition. An n× n matrix A is called nonsingular, or
invertible if there exists an n× n matrix B such that
AB = BA = I_{n}.
Such B is called an inverse of A, and we write B = A^{−1}. Thus we have
Theorem 1. If A and B are both invertible then AB is also invertible and
Theorem 2. If A is invertible then A^{−1} is also invertible and
Theorem 3. If A is invertible then A^{T} is also invertible and
2.3 Linear systems and Inverses
If A is an invertible n× n matrix, then the linear system Ax = b has the unique
solution x = A^{−1}b
Examples. Suppose is invertible 1. Prove that
2. Solve the system equation without using methods of elimination
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