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|
|3. A+ O = A, where O = 
is the zero matrix .
|a+0 = a.||Additive identity|
|4. A+(−A) = O, where −A
= [−aij ] 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
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|
1.4 Properties of transpose
Let r ∈ R and A, B be appropriate matrices
2 Special types of Matrices
2.1 Square matrices
Definition. Let A = [aij ] be a square matrix of order n (n× n type), we say
1. A is a upper triangular matrix if aij = 0 for i > j.
2. A is a lower triangular matrix if aij = 0 for i < j.
3. A is a symmetric if AT = A and skew symmetric if AT = −A.
4. A is a diagonal matrix if aij = 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., aij = 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 In 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, Ap by
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 = In.
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 AT 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−1b
Examples. Suppose is invertible 1. Prove that
2. Solve the system equation without using methods of elimination