 # matrix operations

1 Algebraic properties of matrix operations

 Let A, B, C be m× n matrices Let a ,b, c be any numbers Associative Commutative 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 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 = [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 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 = 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 Prev Next