True/ False

Question 1

For a matrix A the (i, j)-entry of A in the entry in column i and row j. (1a)

Answer 1

F

Question 2

Consider the matrix
(0 0 2)
C= (0 7 0)
(3 0 0)
Then C is diagonal with diagonal entries 2, 7 and 3. (1b)

Answer 2

F

Question 3

Consider the matrix
(4 0 0)
D= (0 7 0)
(0 0 9)
Then D is diagonal with diagonal entries 4, 7 and 9. (1c)

Answer 3

T

Question 4

For all matrices A and B of the same size, A + B = B + A. (1d)

Answer 4

T

Question 5

For all square matrices A and B of the same size, AB = BA (1e)

Answer 5

F

Question 6

The transpose of an m × n matrix is an n × m matrix. (1f)

Answer 6

T

Question 7

The matrices C and D are both symmetric.
(0 0 2)
C= (0 7 0)
(3 0 0)
(4 0 0)
D= (0 7 0)
(0 0 9) (1g)

Answer 7

F

Question 8

If the matrix A is symmetric then the matrix −A is symmetric.(1h)

Answer 8

T

Question 9

If A is an m × n matrix, then the range of the transformation
x → Ax is R^m (2a)

Answer 9

F

Question 10

The range of the transformation x → Ax is the set of all linear combinations of the columns of A (2b)

Answer 10

T

Question 11

If A, B and C are n × n matrices and AB = AC then B = C. (2c)

Answer 11

F

Question 12

If A and B are n × n matrices and AB = I then B = A^−1. (2d)

Answer 12

T

Question 13

If A is an invertible matrix then the inverse of A can be written as 1/A. (2e)

Answer 13

F

Question 14

If A is invertible then the system Ax = b has a unique solution. (2f)

Answer 14

T

Question 15

Let
A = (a b)
(c d).
Then A is invertible if and only if ad-bc > 0. (2g)

Answer 15

F

Question 16

If A and B are invertible then AB is invertible and (AB)^−1 =A^−1B^−1. (2h)

Answer 16

F
If A and B are invertible matrices of the same size, then AB is invertible and
(AB)^−1 = B^−1A^−1.

Question 17

If A is invertible then A^T
is invertible, and the inverse of A^T is the
transpose of A^−1

Answer 17

T

Question 18

Let A be an invertible matrix. Then the matrix obtained by adding
twice the first row to the second may not be invertible. (2j)

Answer 18

F

Question 19

Let A be a non-invertible n × n matrix. Then the linear system
Ax = 0 has infinitely many solutions. (2k)

Answer 19

T

Question 20

A square matrix A is invertible if and only if its reduced row echelon form is the identity. (2l)

Answer 20

T

Question 21

Performing an elementary row operation is equivalent to left multiplication by an elementary matrix. (3a)

Answer 21

T

Question 22

Every invertible matrix A can be expressed uniquely as a product
of elementary matrices. (3b)

Answer 22

F

Question 23

Let A be a non-invertible n × n matrix. Then the matrix obtained
by swapping rows one and two of A is also non-invertible. (3c)

Answer 23

T

Question 24

For matrices A, B and C, if A is row equivalent to B and B is row equivalent to C then A is row equivalent to C. (3d)

Answer 24

T
two matrices are row equivalent if one can be changed to the other by a sequence of elementary row operations

Question 25

Every elementary matrix is invertible. (3e)

Answer 25

T By Lemma 2.5.2: Each elementary matrix is invertible, and its inverse is an elementary matrix of the same type.

Question 26

The product of two n × n elementary matrices must also be an elementary matrix. (3f)

Answer 26

F

Question 27

If E1 and E2 are both n × n elementary matrices then E1E2 = E2E1. (3g)

Answer 27

T

Question 28

A square matrix A is invertible if and only if its reduced row echelon form is the identity. (3h)

Answer 28

T

Question 29

Let A = (1 1) (0 1). Then T_A is the function that assigns to each vector x=(x1,x2) ∈ R2 the vector y = (x1 + x2, x2) ∈ R2 (3i)

Answer 29

T

Question 30

If T : Rn → Rm is linear then T(2u) = T(u) + T(u). (3j)

Answer 30

T

Question 31

For all linear transformations T : Rn → Rn and S : Rn → Rn, we have S ◦ T = T ◦ S. (3k)

Answer 31

F

Question 32

Let A and B be two n × n matrices over R. If AB = BA then TA ◦ TB = TB ◦ TA. (3l)

Answer 32

T

Question 33

Let T : Rn → Rn be an invertible linear transformation. Then T^−1: Rn → Rn is linear. (3m)

Answer 33

T

Question 34

A function T : Rn → Rm is a linear transformation if and only if it is a matrix transformation. (3n)

Answer 34

T