Matrix Flashcards
(13 cards)
Matrix Size Order
Rows x Columns
Matrix Multiplication Rule
A Columns = B Rows
Symmetric v Antisymmetric Matrix
Symmetric if A=A^T
Anti if A=-A^T
Matrix vs Augmented vs Coefficient Form
Matrix has whole Ax=B
Augmented drops x, and has A|B
Coefficient is just the A
Row Echelon + Reduced Row Echelon
a) Non-Zero rows at the bottom
b) First non-zero entry is to the right of the pivot above it
If all pivots are 0, it’s reduced row echelon
Infinite or No Solutions with Matrices
No Solutions if there is a zero row, that has a non-zero answer -> you can’t have 0x+0y=2
Infinite Solutions if there is a zero row with a zero answer, therefore there is x variables and (x-1) equations
Linear Dependence
When one equation is a scalar multiple of another equation
Matrix Rank
How many of the rows are unique - linearly dependent
Homogenous System
Let [A|B], where all B is 0
Will always have the trivial solution, where all of x is equal to 0
Inverse Matrix (What we can do with it & how to find it)
AA^-1=Identity and A^-1A = Identity
We can also use it to find solutions:
Identity*X=A^-1 * B
To find it for 2x2: use the formula
for >3x3, use the Gauss Jordan
How to Calculate the Determinant
for n=2 use the formula
for n>3, do it recursively across any column or row
note the sign matrix
What does the Det mean
if det A = 0, then A has no inverse
if det A != 0, then A has an inverse
If no inverse, system has no solutions
Cramers Rule
Another method to find the solution with a matrix
Define Ai as the Matrix A with the ith column replaced by Bi
Then x=[x1,x2,x3] has the solution xi=det Ai/det A
