Matrices

Math 115 Notes §

The m x n matrix with all zero entries is called a zero matrix, denoted by 0, or simply by 0 if the size is clear

[! Fundamental Properties of Matrix Algebra]

Let A, B, C and let c, d, . We have

M1. A + B | is closed under addition

M2. A + B = B + A | addition is commutative

M3. (A + B) + C = A + (B + C) | addition is associative

M4. cA | is closed under scalar multiplication

M5. c(dA) = (cd)A | Scalar multiplication is associative

M6. (c + d)A = cA + dA | distributive law

M7. c(A+B) = cA + cB | distributive law

Let A . The transpose of A, denoted by A, is the n x m matrix satisfying (A) = (A)

[! Properties of Transpose]

Let A, B and c . Then

a) A

b) (A = A

c) (A + B) = A + B

d) (cA) = cA

Notation §

• A = reference to matrice
• a = element
• a = refers to element in matrice A 1st row 2nd column
• a = refers to element in matrice A 2nd row 3rd column

Rules §

When calculating size of matrix it goes row x column

In this direction

Types of Matrices §

Zero Matrix:

All positions (elements) are 0

Identity Matrix:

Has 1 in diagonals and everything else is 0

Matrix Operations §

Matrix Addition Matrix Subtraction Matrix Multiplication Matrix Determinants Matrix Inverse