True/False Chapter 2

Question 1

If A and B are 2x2 with columns a1, a2 and b1, b2, respectively, then AB=[a1 b1 a2 b2]

Answer 1

false

Question 2

Each column of AB is a linear combination of the columns of B using weights from the corresponding columns of A

Answer 2

false : the roles of A and B should be reversed in the second half statement

Question 3

AB + AC = A(B+C)

Answer 3

true

Question 4

A^T + B^T = (A+B)^T

Answer 4

true

Question 5

The transpose of a product of matrices equals the product of their transposes in the same order

Answer 5

false : the phrase ‘in the same order’ should be ‘in the reverse order’

Question 6

If A and B are 3x3 and B=[b1 b2 b3], then AB=[Ab1+Ab2+ab3]

Answer 6

false : AB must be a 3x3 matrix, but the formula for AB implies that it is 3x1. The plus sign should just be spaces (between columns)

Question 7

The second row of AB is the second row of A multiplied on the right by B

Answer 7

true

Question 8

(AB)C = (AC)B

Answer 8

false : the left-to-right order of B and C cannot be changed, in general

Question 9

(AB)^T = A^T B^T

Answer 9

false

Question 10

The transpose of a sum of matrices equals the sum of their transposes

Answer 10

true

Question 11

A product of invertible nxn matrices is invertible, and the inverse of the product is the product of their inverses in the same order

Answer 11

false : the product matrix is invertible, but the product of inverses should be in the reverse order

Question 12

If A is invertible, then the inverse of A^-1 is A itself

Answer 12

true

Question 13

If A = a b and ad=bc then A is not invertible

c d

Answer 13

true

Question 14

If A can be row reduced to the identity matrix, then A must be invertible

Answer 14

true

Question 15

If A is invertible, then elementary row operations that reduce A to the identity In also reduce A^-1 to In

Answer 15

false

Question 16

In order for a matrix B to be the inverse of A, both equations AB = I and BA = I must be true

Answer 16

true

Question 17

If A and B are nxn and invertible, then A^-1B^-1 is the inverse of AB

Answer 17

false

Question 18

If A = a b and ab-cd /= 0, then A is invertible

c d

Answer 18

false : it’s ad-bc /=0

Question 19

If A is an invertible nxn matrix, then the equation Ax=b is consistent for each b in Rn

Answer 19

true

Question 20

Each elementary matrix is invertible

Answer 20

true

Question 21

If the equation Ax=0 has only the trivial solution, then A is row equivalent to the nxn identity matrix

Answer 21

true : by IMT, if statement d is true, then so is b

Question 22

If the columns of A span Rn, then the columns are linearly independent

Answer 22

true : by IMT

Question 23

If A is an nxn matrix, then the equation Ax=b has at least one solution for each b in Rn

Answer 23

false : only true for invertible matrices

Question 24

If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions

Answer 24

true

Question 25

If A^T is not invertible, then A is not invertible

Answer 25

true

Question 26

If there is an nxn matrix D such that AD = I, then there is also an nxn matrix C such that CA = I

Answer 26

true

Question 27

If the columns of A are linearly independent, then the columns of A span Rn

Answer 27

true

Question 28

If the equation Ax=b has at least one solution for each b in Rn, then the solution is unique for each b

Answer 28

true

Question 29

If the linear transformation (x)–>Ax maps Rn into Rn, then A has n pivot positions

Answer 29

false : the first part of the statement is not part i of the IMT. In fact, if A is any nxn matrix, the linear transformation x–>Ax maps Rn into Rn, yet not every such matrix has n pivot positions

Question 30

If there is a b(vector) in Rn such that the equation Ax=b is inconsistent, then the transformation x–>Ax is not one-to-one

Answer 30

true