Equality
Definition
two matrices A=(aij) an (mxn) matrix and B=(bij) an (rxs) matrix are equal if and only if m-r = n-s = aij-bij = 0 for all i and j
Addition
Definition
let A = (aij) and B = (bij) A+B = C = (aij+bij) -this is only defined if both A and B are (mxn) matrices
Scalar Multiplication
Definition
let A=(aij)
then kA = (kaij)
Matrix Multiplication
Definition
let A=(aij) (mxn) and B=(bij) (nxp) then: AB = C = (cij) where cij = k=1->nΣ(aik)(bkj) -this is only defined if the number of columns in A is the same as the number of rows in B -C is an (mxp) matrix
Is matrix multiplication commutative?
No
in general AB≠BA
if AB=BA then both A and B must be square matrices
Square Matrix
Definition
an (nxn) matrix is square
Transpose
Definition
if we interchange the rows and columns of a matrix A=(aij) (mxn) then we obtain the transpose, A^T=(aji) (nxm)
Transpose and Matrix Multiplication
Let A be an (nxm) matrix and B be a (mxp) matrix, then
(AB)^T = A^T*B^T
Identity Matrix
Definition
an identity matrix I=(iij) (nxn) is an all zero matrix except for the entries I(aa)=1
such that for an (mxn) matrix B:
ImB = MIn = B
Inverse
Definition
Let A and B be two (nxn) matrices, if AB = In = BA, then B is the multiplicative inverse of A
Inverse matrices only exist for square matrices
Invertible/Non-Singular
Definition
If a square matrix A has an inverse ten we say that A is invertible or non-singular
If a matrix A is invertible then its inverse is unique
Non-Invertible/Singular
Definition
If a square matrix A does not have an inverse then A is non-invertible or singular
Inverse and Matrix Multiplication
If A and B are invertible (nxn) matrices, then:
i) AB is invertible
ii) (AB)^-1 = B^-1 A^-1
If A is not invertible then AB is also not invertible for any B (nxn)
Transpose and Inverse
Let A be an (nxn) invertible matrix, then A^T is also invertible and:
(A^T)^-1 = (A^-1)^T
Method of Finding Matrix Inverses
1) form an augmented matrix (A|In) where A is an (nxn) matrix
2) apply row operations until a row of zeros appears in the left block and we conclude A has no inverse
OR, the identity matrix appears in the left block and we conclude that A is invertible and its inverse is the matrix in the right block
Inverse of a 2x2 Matrix
let A = (a b) (c d) then A^-1 = 1/(ad-bc)(d -b) (-c a)
Zero Matrices and Inverses
If A is an (nxn) matrix consisting entirely of zeros, a zero matrix, then A is not invertible
Elementary Matrix
Definition
- a matrix obtained by applying an elementary row operation to an identity matrix In
- by definition all elementary matrices area square
- applying an elementary row operation to an arbitrary matrix A is the same as multiplying on the left by the corresponding matrix
Elementary Matrices and Their Inverses
- all elementary row operations are invertible
- the inverse if an elementary matrix corresponds to multiplying on the left by the opposite row operation
Inverses and Elementary Matrices
If A is an (nxn) matrix whose reduced row echelon form is the identity matrix In, then A is invertible
AND
A and A^-1 are both products of elementary matrices
Conversely if A is an invertible (nxn) matrix then its RREF is In
When does the linear system Ax=b have a unique solution?
If A is an (nxn) matrix and b is a column vector, then
Ax = b
has a unique solution if and only if A is invertible