Deck 3: Matrices

Question 1

Let
and find
and
where

Answer 1

Question 2

Let

, and find
and
where

Answer 2

Question 3

Determine if the given vector is in the range of
where

Answer 3

Yes,

is in the range of T.

Question 4

Determine if the given vector is in the range of
where

Question 5

Find

, given that

Question 6

Identify the matrix
such that
where

Question 7

Determine whether
is one-to-one, and whether
is onto, where
and

Question 8

Determine whether
is one-to-one, and whether
is onto, where
and

Question 9

Determine whether
is one-to-one, and whether
is onto, where
and

Question 10

Suppose that
for

. Sketch a graph of the image under
of the unit square in the first quadrant of R².

Question 11

If
is a linear transformation, then
for all vectors
and

.

Question 12

If
is a linear transformation, then
for all vectors
and

.

Question 13

If
is a one-to-one linear transformation from R² to R², then the image of the unit square under
is a parallelogram with positive area.

Question 14

If
for a matrix

, and
is a one-to-one linear transformation, then the number of columns of
is less than or equal to the number of rows of

.

Question 15

If
for a matrix

, and
is an onto linear transformation, then the number of columns of
is greater than or equal to the number of rows of

.

Question 16

Evaluate AB, where

Question 17

Evaluate AB-BA, where

Question 18

Evaluate 3AB, where

Question 19

Evaluate AA^T-A^TA, where

Question 20

Evaluate
where
is a positive integer, and

Question 21

Solve for the scalars a, b, c in the following equation:

Question 22

Let
and
be the linear transformations given by

Find the matrix
such that

.

Question 23

Let
and
be the linear transformations given by

Find the matrix
such that

, and the matrix
such that

.

Question 24

Suppose
is a
matrix, and
is a
matrix, and the product AB is defined and is a
matrix. Determine

,

, and

.

Question 25

Determine AB using block multiplication, where

Question 26

If
is any square matrix, then the matrix
is symmetric.

Question 27

If
is any square matrix, then the matrix
is symmetric.

Question 28

If
and
are
symmetric matrices, then AB is an
symmetric matrix.

Question 29

If
and
are
symmetric matrices, then
is an
symmetric matrix.

Question 30

If
is an
upper triangular matrix, and
is an
lower triangular matrix, then AB^T is an
upper triangular matrix.

Question 31

If
is any matrix, then the matrices
and AA^T are both symmetric.

Question 32

Use the quick formula to determine the inverse of

Question 33

Use an augmented matrix and row operations to find the inverse of

Question 34

Use an augmented matrix and row operations to find the inverse of

Question 35

Use an augmented matrix and row operations to find the inverse of

Question 36

Use an augmented matrix and row operations to find the inverse of

Question 37

Use the matrix
to find the solutions to the linear system

Question 38

Use the matrix
to find the solutions to the linear system

Question 39

Determine if the given linear transformation
is invertible, and if so, find

.

Question 40

Determine if the given linear transformation
is invertible, and if so, find

.

Question 41

Determine if the given linear transformation
is invertible, and if so, find

.

Question 42

If
is a square matrix such that
has a solution for some particular vector

, then
is invertible.

Question 43

If
and
are
invertible matrices, then so is

Question 44

If
is a square invertible matrix, then
is an invertible matrix for every positive integer

, with

Question 45

If
is a square invertible matrix, and
then

Question 46

If
and
are
matrices, and
and
are both not invertible, then
is not invertible.

Question 47

Use the given LU factorization to solve
where

Question 48

Use the given LU factorization to solve
where

Question 49

Find an LU factorization of

Question 50

Find an LU factorization of

Question 51

Find an LU factorization of

Question 52

Find an LU factorization of

Question 53

Find an LDU factorization of

Question 54

Find an LDU factorization of

Question 55

Determine
by evaluating
from the LDU factorization

Question 56

Every matrix
has an LU factorization.

Question 57

If
is a square invertible matrix which has an LU factorization, then the factorization is unique.

Question 58

Every matrix
has an LDU factorization.

Question 59

If
is a square matrix which has an LU factorization, then
is invertible.

Question 60

Every elementary matrix has an LU factorization.

Question 61

If both A and B are
matrices such that their product AB has an LU factorization, then both A and B have LU factorizations.

Question 62

Determine if the matrix is a stochastic matrix.

Question 63

Determine if the matrix is a stochastic matrix.

Question 64

Find the state vector
for the given stochastic matrix and initial state vector.

Question 65

Find the state vector
for the given stochastic matrix and initial state vector.

Question 66

Find all steady-state vectors for the given stochastic matrix.

Question 67

Find all steady-state vectors for the given stochastic matrix.

Question 68

Find all steady-state vectors for the given stochastic matrix.

Question 69

Find all steady-state vectors for the given stochastic matrix.

Question 70

Determine if the given matrix is regular.

Question 71

Determine if the given matrix is regular.

Question 72

If
and
are
stochastic matrices, then
is a stochastic matrix.

Question 73

A matrix
is doubly stochastic if and only if
is doubly stochastic.

Question 74

The
identity matrix
is a regular stochastic matrix.

Question 75

If
and
are
doubly stochastic matrices, then AB is a doubly stochastic matrix.

Question 76

If
and
are
regular stochastic matrices, then AB is a regular stochastic matrix.