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Flashcards in Chapter 2 Deck (21)
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1
Q

Equality

Definition

A

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

2
Q

Addition

Definition

A
let A = (aij) and B = (bij)
A+B = C = (aij+bij)
-this is only defined if both A and B are (mxn) matrices
3
Q

Scalar Multiplication

Definition

A

let A=(aij)

then kA = (kaij)

4
Q

Matrix Multiplication

Definition

A
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
5
Q

Is matrix multiplication commutative?

A

No
in general AB≠BA
if AB=BA then both A and B must be square matrices

6
Q

Square Matrix

Definition

A

an (nxn) matrix is square

7
Q

Transpose

Definition

A

if we interchange the rows and columns of a matrix A=(aij) (mxn) then we obtain the transpose, A^T=(aji) (nxm)

8
Q

Transpose and Matrix Multiplication

A

Let A be an (nxm) matrix and B be a (mxp) matrix, then

(AB)^T = A^T*B^T

9
Q

Identity Matrix

Definition

A

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

10
Q

Inverse

Definition

A

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

11
Q

Invertible/Non-Singular

Definition

A

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

12
Q

Non-Invertible/Singular

Definition

A

If a square matrix A does not have an inverse then A is non-invertible or singular

13
Q

Inverse and Matrix Multiplication

A

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)

14
Q

Transpose and Inverse

A

Let A be an (nxn) invertible matrix, then A^T is also invertible and:
(A^T)^-1 = (A^-1)^T

15
Q

Method of Finding Matrix Inverses

A

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

16
Q

Inverse of a 2x2 Matrix

A
let A = (a   b)
            (c   d)
then 
A^-1 = 1/(ad-bc)(d  -b)
                         (-c  a)
17
Q

Zero Matrices and Inverses

A

If A is an (nxn) matrix consisting entirely of zeros, a zero matrix, then A is not invertible

18
Q

Elementary Matrix

Definition

A
  • 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
19
Q

Elementary Matrices and Their Inverses

A
  • all elementary row operations are invertible

- the inverse if an elementary matrix corresponds to multiplying on the left by the opposite row operation

20
Q

Inverses and Elementary Matrices

A

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

21
Q

When does the linear system Ax=b have a unique solution?

A

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