quiz 4

Question 1

what is the determinant of a 2x2 matrix?

Answer 1

ad - bc

Question 2

what does theorem 4 state?

Answer 2

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

Question 3

a matrix that is not invertible is called

Answer 3

a singular matrix

Question 4

a matrix that is invertible is called

Answer 4

a nonsingular matrix

Question 5

a matrix A^-1 is the inverse of A if

Answer 5

AA^-1 = I
and
A^-1A = I

Question 6

what is the formula for the inverse of a 2x2 matrix?

Answer 6

1
———- * [d -b]
ad - bc [-c a]

Question 7

what does theorem 5 state?

Answer 7

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

Question 8

how many parts are there in theorem 6?

Answer 8

3

Question 9

what does theorem 6 state?

Answer 9

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

Question 10

what is theorem 6 in easier terms?

Answer 10

a) the inverse of the inverse of A is A

b) (AB)^-1 = B^-1 * A^-1

c) (A^T)^-1 = (A^-1)^T

Question 11

what is part a of theorem 6?

Answer 11

(A^-1)^-1 = A

Question 12

what is part b of theorem 6?

Answer 12

(AB)^-1 = B^-1 * A^-1

Question 13

what is part c of theorem 6?

Answer 13

(A^T)^-1 = (A^-1)^T

Question 14

what does theorem 7 state?

Answer 14

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

Question 15

what is the algorithm for finding A^-1?

Answer 15

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.

Question 16

what is another name for theorem 8?

Answer 16

the invertible matrix theorem

Question 17

what is the first part of the invertible matrix theorem?

Answer 17

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

Question 18

what are the statements in theorem 8?

Answer 18

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

Question 19

what are parts a - c of theorem 8?

Answer 19

a) A is an invertible matrix
b) A is row equivalent to I
c) A has n pivot positions

Question 20

what are parts d - f of the invertible matrix theorem?

Answer 20

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

Question 22

what are parts g - i of theorem 8?

Answer 22

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

Question 23

what are parts j - l of the invertible matrix theorem?

Answer 23

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