Linear

If a matrix is invertible, is the inverse unique? Explain or show a proof.

A and B are symmetric matrices. Is AB symmetric

No

If AB is invertible, then is A and B also invertible?

Yes, vice versa

(AA)² is invertible. Is A invertible?

Yes

Dependent vs independent equation

Independent = rank

Just an equation with an answer

Dependent = # of equ - rank

Scalar mutiple of another independent equation which reduces to all 0s

If a matrix is not row equivalent to another, can they have the same solutions

Yes: no solution

Given an equation with variables in the A and b vector, how do u find: (a and b)

Unique sol

Infinitely many sol

No sol

First, reorder the variables using row equivalence and simplify to make the variables at the last of the REF

1 unique:

a cannot = 0

b all real numbers

Infinite

a = 0

b = 0

a = 0

b cannot = 0

Matrix multiplications are..

Associative

Distributive

Not commutative

Zero matrix

Diagonal

Off diagonal

Identity

Triangular

Square

Symmetric

Skew

All 0

# on diagonal

# not on diagonal (0 on diagonal)

1 on diagonal (0 on not)

UT: REF

LT: upside down REF

n=m

Square but matrix = transposed

Square but matrix = -ve transposed