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)
tr(A+B) = tr(A) + tr(B)
tr(cA) = c tr(A)
tr(AB) = tr(BA)
TRANSPOSES
(A ) = A
T T T
(A+B) =A + B
(kA) = kA T
T T T
(AB) = B A
(A )-1 = (A ) T
T T
A A and AA are symmetric, and if A is invertible, they are also invertible INVERSES
-1 -1 -1
(AB) = A B
(A ) = A = (A )1 n
-1 -1
(A ) = A
(kA) = k A -1
A diagonal matrix is invertible only if all of its diagonal entries are non-zero
A matrix is invertible as long as its determinant isn’t zero
If A and B are square, same dimensions, and AB is invertible, A and B are each invertible
If A is a singular, square matrix, Ax=0 has infinitely many solutions
If A is invertible, and the first row is added to the second, A is still invertible
DETERMINANTS
A is invertible only if det(A) isn’t 0
-1
det(A ) = 1/det(A)
det(cA) = c^n(det(A))
det(A^n) = (det(A))^n
T
det(A) = det(A )
det(AB) = det(A)det(B)
T T
det(A+B ) = det(A + B)
det(A +B ) = det(A +B)
TRIANGULAR MATRICES
Transpose
o Transpose LOWER = UPPER
o Transpose UPPER = LOWER
multiply
o product of LOWERS = LOWER
o product of UPPERS = UPPER
Inverse
o Inverse of LOWER = LOWER
o Inverse of UPPER = UPPER
SYMMETRY
If A and B are nxn and symmetric
o A+B is symmetric
o A-B is symmetric
o kA is symmetric
AB is symmetric ONLY if the matrices COMMUTE
-1
If A is symmetric, invertible, A is symmetric T T T T
if A A and AA are symmetric, and A is invertible, A A and AA are also invertible
the product of SKEW SYMMETRIC matrices that commute is symmetric
If A and B are symmetric, AB + BA is sy

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