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MATH 1ZC3 Study Guide - Midterm Guide: Diagonal Matrix, Symmetric Matrix, Transpose


Department
Mathematics
Course Code
MATH 1ZC3
Professor
Chris Mc Lean
Study Guide
Midterm

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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
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