Chapter 2 Definitions Flashcards
Definition of Equality of Matrices
equal when same size and aij = bij for 1
Definition of Matrix Addition
A + B = [aij + bij]
Definition of Scalar Multiplication
cA = [caij]
Definition of Matrix Multiplication
AB = [cij] where cij = sum(k=1, n, aikbkj)
Linear Combinations of Column Vectors
x11a1 + x21a2 + … + xnan = b, where A is matrix mxn, x is nx1, b nx1
Commutative property of addition
A + B = B + A
associative property of addition
A + (B + C) = (A + B) + C
associative property of multiplication
(cd)A = c(dA)
multiplicative identity
1A = A
distributive property
c(A + B) = cA + cB
distributive property
(c +d)A = cA + dA
property of zero matrix
A + Omn = A
property of zero matrix
A + (-A) = Omn
property of zero matrix
if cA = Omn then c=0 or A = Omn
associative property of matrix multiplication
A(BC) = (AB)C
distributive property of matrix multiplication
A(B+C) = AB + AC
distributive property of matrix multiplication
(A+B)C = AC + BC
scalar properties of matrix multiplication
c(AB)=(cA)B=A(cB)
property of identity matrix
AIn = A
property of identity matrix
ImA = A
properties of transposes
(A^T)^T=A
properties of transposes
(A+B)^T=A^T+B^T
properties of transposes
(cA)^T=c(A^T)
properties of transposes
(AB)^T=B^T*A^T
Definition of the Inverse of a Matrix
AB = BA = In
Properties of Inverse Matrices: 1
(A^-1)^-1=A
Properties of Inverse Matrices: 2
(A^k)^-1 = (A^-1)^k
Properties of Inverse Matrices: 3
(cA)^-1=c^-1A^-1
Properties of Inverse Matrices: 4
(A^T)^-1=(A^-1)^T
Inverse of a Product
(AB)^-1=B^-1*A^-1
Definition of an Elementary Matrix
nxn where it can be obtained from the identity matrix In by a single elementary row operation
Invertible Matrix Theroem: 1
A is invertible if A is an nxn matrix
Invertible Matrix Theroem: 2
Ax=b has a unique solution for every nx1 column matrix b
Invertible Matrix Theroem: 3
Ax=0 has only the trivial solution
Invertible Matrix Theroem: 4
A is row-equivalent to I
Invertible Matrix Theroem: 5
A can be written as the product of elementary matrices
