Matrices

Math 115 Notes §

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

[! Fundamental Properties of Matrix Algebra]

Let A, B, C $\in M_{m\cdot n}(\mathbb{R})$ and let c, d, $\in \mathbb{R}$. We have

M1. A + B $\in M_{m\cdot n}(\mathbb{R})$ | $M_{m\cdot n}$ is closed under addition

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

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

M4. cA $\in M_{m\cdot n}(\mathbb{R})$ | $M_{m\cdot n}(\mathbb{R})$ 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 $\in M_{m\cdot n}(\mathbb{R})$. The transpose of A, denoted by A${}^T$, is the n x m matrix satisfying (A${}^T$)${}_{ij}$ = (A)${}_{ji}$

[! Properties of Transpose]

Let A, B $\in M_{m\cdot n}(\mathbb{R})$ and c $\in \mathbb{R}$. Then

a) A${}^T$ $\in M_{n\cdot m}(\mathbb{R})$

b) (A${}^T{)}^T$ = A

c) (A + B)${}^T$ = A${}^T$ + B${}^T$

d) (cA)${}^T$ = cA${}^T$

Notation §

• A = reference to matrice
• a = element
• a${}_{12}$ = refers to element in matrice A 1st row 2nd column
• a${}_{23}$ = refers to element in matrice A 2nd row 3rd column

Rules §

When calculating size of matrix it goes row x column

In this direction $\Rightarrow$ $\Downarrow$

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