2.1, 2.2, 2.3 Flashcards
(21 cards)
let A,B,C be n*n matrices. r and s are scalars A+B = (A+B) + C = A + 0 = r(A+B) = (r+s)A = r(sA)=
B+A A+(B+C) A rA+rB rA+sA (rs)A
in general AB does not equal
BA
if AB = AC , then it is not true that
B = C
fi the product of AB is the zero matrix, you can not conclude that A =
B =
0
let A and B denote matrices whose sizes are not appropriate for the following sums and products (A^t) = (A+B)^t = for any scalar r (rA)^t = (AB)^t =
A
A^t +B^t
rA^t
B^tA^t
the transpose of a product of matrices equals
the product of their transposes in reverse
an nn matrix A is invertible if there is an nn matrix C such that CA= I and ….
and AC = I
matrix that is not invertible
singular matrix
invertible matrix
nonsingular matrix
square n by n matrix whose non diagonal entries are zero
diagonal matrix.
if ad-bc does not equal 0 then
the matrix is invertible
ad-bc equals the
determinant of A
det a =
ad-bc
if A is an invertible n by n matrix, then for each b in R^n, the equation Ax = b has the unique solution
x = A^-1b
if A is an invertible matrix, then Ainverse is invertible and
the inverse of A-inverse is A
if A and B are n by n invertible matrices, then so is AB and the inverse of AB is the
product of the inverses of A and B in reverse order
if A is an invertible matrix, then so is A^t, and the inverse of A^t is
the transpose of A-inverse
obtained by performing a single elementary row operation Onan identity matrix
elementary matrix
an n by n matrix A is invertible if and only if A is row equivalent to In. any sequence of elementary operations that reduces A to In also transforms
In to A-inverse
A is an n by n matrix. the following statements are equivalent. All are truee or all are false A is an A is row equivalent to the A has _ pivot positions the equation Ax= 0 has only the the columns of A form a the linear transformation x to Ax is the equation Ax = b has the columns of A span the linear transformation x to Ax maps there is an n by n matrix C such that there is an n by n matrix D such that A^t is an \_\_\_\_ matrix
invertible matrix n by n identity matrix n pivot positions only the trivial solutions form a linearly independent set is one to one at least one solution for each b in R^n R^n maps R^n onto R^n CA = I AD = I invertible matrix
x(A) = Ax so Ax(A-inverse) =
x