quiz 4 Flashcards
what is the determinant of a 2x2 matrix?
ad - bc
what does theorem 4 state?
let A = [a b], [c d] if ad - bc (the determinant) does not = 0, then A is invertible and A^-1 = 1/(ad-bc)[[d -b], [-c a]]. if ad - bc = 0, then A is not invertible
a matrix that is not invertible is called
a singular matrix
a matrix that is invertible is called
a nonsingular matrix
a matrix A^-1 is the inverse of A if
AA^-1 = I
and
A^-1A = I
what is the formula for the inverse of a 2x2 matrix?
1
———- * [d -b]
ad - bc [-c a]
what does theorem 5 state?
if A is an invertible nxn (square) matrix, then for each b in R^n, the equation Ax = b has the unique solution A^-1b
how many parts are there in theorem 6?
3
what does theorem 6 state?
a) if A is an invertible matrix, then A^-1 is invertible and (A^-1)^-1) = A
b) if A and B are nxn (square) matrices, then so is AB, and the inverse of AB is the product of the inverses of A and B in the reverse order. that is, (AB)^-1 = B^-1 * A^-1
c) if A is an invertible matrix, then so is A^T, and the inverse of A^T is the transpose of A^-1. that is, (A^T)^-1 = (A^-1)^T
what is theorem 6 in easier terms?
a) the inverse of the inverse of A is A
b) (AB)^-1 = B^-1 * A^-1
c) (A^T)^-1 = (A^-1)^T
what is part a of theorem 6?
(A^-1)^-1 = A
what is part b of theorem 6?
(AB)^-1 = B^-1 * A^-1
what is part c of theorem 6?
(A^T)^-1 = (A^-1)^T
what does theorem 7 state?
an nxn (square) matrix is invertible if and only if it is row equivalent to I, and in this case, any sequence of elementary row operations that reduces A to I also transforms I to A^-1
what is the algorithm for finding A^-1?
row reduce the augmented matrix [A I]. if A is row equivalent to I, then [A I] is row equivalent to [I A^-1]. otherwise, A does not have an inverse.
what is another name for theorem 8?
the invertible matrix theorem
what is the first part of the invertible matrix theorem?
let A be a square nxn matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false
what are the statements in theorem 8?
a) A is an invertible matrix
b) A is row equivalent to I
c) A has n pivot positions
d) the equation Ax = 0 has only the trivial solution
e) the columns of a form a linearly independent set
f) the linear transformation x –> Ax is a one-to-one
g) the equation Ax = b has at least one solution for each b in R^n
h) the columns of A span R^n
i) the linear transformation x –> Ax maps R^n onto R^n
j) there is an nxn matrix C such that CA = I
k) there is an nxn matrix D such that AD = I
l) A^T is an invertible matrix
what are parts a - c of theorem 8?
a) A is an invertible matrix
b) A is row equivalent to I
c) A has n pivot positions
what are parts d - f of the invertible matrix theorem?
d) the equation Ax = 0 has only the trivial solution
e) the columns of a form a linearly independent set
f) the linear transformation x –> Ax is a one-to-one
what are parts g - i of theorem 8?
g) the equation Ax = b has at least one solution for each b in R^n
h) the columns of A span R^n
i) the linear transformation x –> Ax maps R^n onto R^n
what are parts j - l of the invertible matrix theorem?
j) there is an nxn matrix C such that CA = I
k) there is an nxn matrix D such that AD = I
l) A^T is an invertible matrix
