# 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 squarematrices, I is called the idetity. |
a 1 = a | Mult. identity |

5. A^{−1} exits only whenthey 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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