Chapter 1.4 Flashcards
General rule concerning parentheses in matrix arithmetics
Given any sum or any product of matrices, pairs of parenthesis can be inserted or deleted anywhere without affecting the end result.
Properties of matrix arithmatics (9)
1) A + B = B + A (commutative law)
2) A + (B + C) = (A + B) + C (associative law)
3) A(BC) = (AB)C (associative law for multiplication)
4) A(B + C) = AB + AC (left distributive law)
5) (B + C)A = BA + CA (right distributive law)
6) a(B + C) = aB + aC
7) (a + b)C = aC + bC
8) a(bC) = (ab)C
9) a(BC) = (aB)C = B(aC)
Properties of Zero-matrices (5)
1) A + 0 = 0 + A = A
2) A - 0 = A
3) A - A = 0
4) A * 0 = 0 * A = 0
5) if cA = 0 then c = 0 or A = 0
If R is the reduced row echelon form of a square matrix A, then one of which two facts are true?
1) R has at least one row of zeros
or 2) R is the identity matrix
What is the definition of an invertible matrix
If A is a square matrix, and if a matrix B of the same size can be found such that AB = BA = I, then A is said to be invertible (or nonsingular) and B is called the inverse of A. If no such matrix B can be found, then A is said to be singular.
If B and C are both inverses of A then
B = C
A =
a b
c d
Is invertible if
ad =/= bc
If A =
a b
c d
And is invertible, then the inverse of A is
1/(ad - bc) *
d -b
-c a
Ithe determinant of the 2x2 matix A is
ad - bc, denoted det(A)
If A and B are invertible matrices of the same size, then inv(AB) is
inv(B) * inv(A)
If A is invertible and n is a nonnegative integer, then (3)
1) inv(A) is invertible and inv(inv(A)) = A
2) A^n is invertible and inv(A^n) =A^-n = inv(A)^n
3) kA is invertible and inv(kA) = (1/k)*(inv(A)
Properties of transposes
1) tra(tra(A)) = A
2) tra(A + B) = tra(A) + tra(B)
3) tra(kA) = ktra(A)
4) tra(BA) = tra(A)tra(B)
If A is an invertible matrix, then what can be said of its transpose
tra(A) is also invertible and
inv(tra(A)) = tra(inv(A))
What can be said about a square matrix with a row or column of zeros
It is singular and has no inverse
