A matrix is a rectangular array of Numbers
Operations
Matrix Addition
- Add elementwise
- Operand matrices must have the same dimensions
Matrix Multiplication
Let
- Then, the
entry of is , where we are performing many Dot Products to calculate the matrix product - Then
Multiplication Properties
Transpose
Transpose Properties
Inverse
- Row reduce
until you get for A - Keep track of the Elementary Matrix for each Row Operation
- The product of all of the elementary matrices is the inverse of
, by definition
may not be invertible. In this case, you won’t be able to row reduce it to the Identity Matrix
Invertible
You cannot always take the inverse of a matrix. Whether or not it is possible is called Invertibility. See also Invertible Matrix Theorem, Invertibility Theorem.
Invertibility Definition:
- A is invertible
A is square - A is invertible
it is row equivalent to the identity - Linearly dependent
Singular
- Mnemonic: After the trial, Johnny Depp was Single
- Linearly independent
Invertible
- This is just the inverse of the above
is invertible - Basically means that A is Bijective
invertible