School

McMaster UniversityDepartment

MathematicsCourse Code

MATH 1ZC3Professor

Chris Mc LeanStudy Guide

MidtermThis

**preview**shows page 1. to view the full**4 pages of the document.**HOMOGENEOUS SYSTEMS

if all equations are homogeneous, matrix must be consistent

o All have the trivial solution (x1 = 0, x2 = 0, …)

o They may have additional solutions as well

in this case, they have infinitely many solutions

o If system has more unknowns than equations, it has infinitely many solutions

thus has a non-trivial solution

If a homogeneous linear system has n unknowns, and r non-zero rows, it has n-r free variables

MATRIX ADDITION

A+B=B+A

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

MATRIX MULTIPLICATION

A(BC) = (AB)C

A(B+C) = AB+AC

(B-C)A = BA-CA

a(B+C) = aB+ aC

(a+b)C = ac+bc

a(bC) = (ab)C

a(BC) =(aB)C = B(aC)

TRACES

The trace of A is undefined if A is not square

The trace is the sum of entries on the principal diagonal

tr A = tr(A)T

tr(A+B) = tr(A) + tr(B)

tr(cA) = c tr(A)

tr(AB) = tr(BA)

TRANSPOSES

(AT)T= A

(A+B)T=AT + BT

(kA)T = kAT

(AB)T= BTAT

(AT) -1 = (A-1)T

ATA and AAT are symmetric, and if A is invertible, they are also invertible

###### You're Reading a Preview

Unlock to view full version