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Theorem 1 ch2.1 Sum and Scalar Multiples of Matrices

Let A, B, and C be matrices of the same size, and let r and s be scalars

a. A + B = B + A

b. (A + B) + C = A + (B + C)

c. A + 0 = A

d. r(A + B) = rA + rB

e. (r + s)A = rA + sA

f. r(sA) = (rs)A

Matrix multiplication

If A is an m x n matrix and B is an n x p matrix then the product is the m x p matrix whose columns are Ab_{1}, ... , Ab_{p}

AB = A[b1 b2 ... bp]

where b is the columns of B

Row-Column Rule for Computing AB

Properties of matrix multiplication theorem ch2.1

let A be an mxn matrix and let B and C have sizes for which the indicated sums and products are defined:

a. A(BC)=(AB)C**associative law of multiplication**

b. A(B+C)= AB + AC**left distributive law**

c. (B+C)A= BA+CA**right distributive law**

d. r(AB)= rA(B)= A(rB)

e. I_{m}A=A=AI_{n} **identity for matrix multiplication**

a. A(BC)=(AB)C

b. A(B+C)= AB + AC

c. (B+C)A= BA+CA

d. r(AB)= rA(B)= A(rB)

e. I

WARNINGS CH2.1

- In general AB =/= BA
- The cancellation laws do NOT hold for matrix multiplication. That is, if AB = AC then it is NOT true in general that B = C
- If a product AB is the zero matrix you CANNOT conclude in general that either A = 0 or B = 0

First row becomes first column, second row becomes second column, etc.

A *m*x*n* -> A^{T} *n*x*m*

[1 2] [1 2 0]

[2 1] -> [2 1 1]

[0 1]

Theorem 2.3: Transpose matrices

Transpose of a product of matrices

def: invertible matrix

singular matrix

Nonsingular Matrix Equivalences

2.4) 2x2 Invertible by Determinant Theorem

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