Chapter 5 - Introduction to Matrix Algebra

matrix

a rectangular array of elements in the form of rays and columns

order

the dimensions of a matrix; m x n if a matrix has m rows and n columns

What are the stipulations for matrix A to be equal to matrix B?

• A and B have the same order

• all corresponding entries are equal

Matrix Addition

Let A and B be m x n matrices. Then A + B is an m x n matrix where (A + B)_ij = a_ij + b_ij. If the orders of A and B are not identical, A + B is not defined

Scalar Multiplication

Let A be an m x n matrix and c a scalar. Then cA is the m x n matrix obtained by multiplying c times each entry of A; that is (cA)_ij = ca_ij

Matrix Multiplication

Let A be an m x n matrix and let be an n x p matrix. The product of A and B, denoted by AB, is an m x p matrix whose ith row jth column entry is:

(AB)_ij = a_i1b_j1 + a_i2b_j2 + … + a_inb_jn

for 1 ≤ i ≤ m and 1 ≤ j ≤ p

What is the order of AB for A_{m x n} and B_{n x p}?

m x p

square matrix

any n x n matrix

0_{m x n}

the notation of m x n matrices whose entries are all 0

AA or A²

the notation of a square matrix A multiplied by itself

diagonal matrix

a square matrix D where d_ij = 0 whener i ≠ j

identity matrix

the n x n diagonal matrix I_n whose diagonal components are all 1’s is called the identity matrix

Matrix Inverse

Let A be an n x n matrix. If there exists an n x n matrix B such that AB = BA = I, then B is a multiplicative inverse of A is denoted by A^-1

True or false: a matrix can have multiple inverses

false; a matrix’s inverse, if it exists, is unique

Determinant of a 2 by 2 matrix

Let A = (^a_c ^b_d). The determinant of a is the number det A = ad - bc

Inverse of 2 by 2 matrix

Let A = (^a_c ^b_d). If det A ≠ 0, then A^-1 = 1/detA (^d_-c ^-b_a)

Commutative Law of Addition

A + B = B + A

Associative Law of Addition

A + (B + C) = (A + B) + C

Distributive Law of a Scalar over Matrices

c(A + B) = cA + cB, where c ∈ ℝ

Distributive Law of Scalars over a Matrix

(c₁ + c₂)A = c₁A + c₂A, where c₁c₂ ∈ ℝ

Associative Law of Scalar Multiplication

c₁(c₂A) = (c₁ × c₂)A where c₁, c₂ ∈ ℝ

Zero Matrix Annihilates all Products

0A = 0, where 0 is the zero matrix

Zero Scalar Annihilates all Products

0A = 0, where 0 on the left side is the scalar zero

Zero Matrix is an identity for Addition

A + 0 = A

Negation produces additive inverses

A + (-1)A = 0

Right Distributive Law of Matrix Multiplication

(B + C)A = BA + CA

Left Distributive Law of Matrix Multiplication

A(B + C) = AB + AC

Associative Law of Multiplication

A(BC) = (AB)C

Identity Matrix is a Multiplicative Identity

IA = A and AI = A

Involution Property of Inverse

if A^-1 exists, (A^-1)^-1 = A

Inverse of Product Rule

if A^-1 and B^-1 exist, (AB)^-1 = B^-1A^-1

Matrix Oddity #1

AB may be different from BA

Matrix Oddity #2

there exist matrices A and b such that AB = 0, and yet A ≠ 0 and B ≠ 0

Matrix Oddity #3

there exists matrices a where A ≠ 0, and yet A² = 0

Matrix Oddity #4

there exists matrices A where A² = A with A ≠ I and A ≠ 0

Matrix Oddity #5

there exists matrices A where A² = I, where A ≠ I and A ≠ -I