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⁻¹b

Examples. Suppose is invertible 1. Prove that

2. Solve the system equation without using methods of elimination