Deck 5: Determinants

Question 1

Find
and
for the matrix

Answer 1

Question 2

Find
and
for the matrix

Answer 2

Question 3

Find
and
for the matrix

Answer 3

Question 4

Find
and
for the matrix

Question 5

Use the Shortcut Method to find the determinant of the matrix

Question 6

Find all values of a such that the matrix A is not invertible.

Question 7

Determine all real values of λ such that
for the matrix

Question 8

Determine all real values of
such that
for the matrix

Question 9

Evaluate

,

,

, and
where

Question 10

Evaluate

,

,

, and
where

​

Question 11

Let A and B be
matrices. Then

.

Question 12

Let A be a
matrix. Then

.

Question 13

If A is a square matrix, and

, then A is invertible.

Question 14

If A is a
matrix and
is a scalar, then

.

Question 15

If A is a
matrix, and B is an
matrix, then

.

Question 16

Compute the determinant of the given matrix A by using row operations to reduce to echelon form.

Question 17

Compute the determinant of the given matrix A by using row operations to reduce to echelon form.

Question 18

Compute the determinant of the given matrix A by using row operations to reduce to echelon form.

Question 19

Suppose the matrix

results from a matrix A using the row operations
and

. Find

.

Question 20

Suppose the matrix

results from a matrix A using the row operations

,

,

, and

. Find

.

Question 21

Suppose A, B, and C are
matrices and

,

, and

. Find each of the following.(a)
(b)
(c)
(d)

Question 22

Partition matrix A to evaluate

, where

Question 23

Partition matrix A to evaluate

, where

Question 24

Use determinants to decide if a unique solution exists for the linear system

Question 25

Use determinants to decide if a unique solution exists for the linear system

Question 26

If a
matrix B is obtained from a matrix A by multiplying every row of A by

, then

.

Question 27

If

, then at least one entry of A has value

.

Question 28

If A and B are square matrices, and

, then either
or

.

Question 29

If A is invertible, then

.

Question 30

If A, B, C, and D are square matrices of the same size, then

.

Question 31

A square matrix
is called nilpotent if
for some positive integer k. Prove that if
is nilpotent, then

.

Question 32

If applicable, use Cramer's Rule to find the unique solution of the given linear system. If Cramer's Rule is not applicable, explain why.

Question 33

If applicable, use Cramer's Rule to find the unique solution of the given linear system. If Cramer's Rule is not applicable, explain why.

Question 34

Without solving for

, use Cramer's Rule to find the value of
in the linear system.

Question 35

Without solving for

, use Cramer's Rule to find the value of
in the linear system.

Question 36

Find
and use it to determine

, where

Question 37

Find
and use it to determine

, where

Question 38

Use determinants to find the area of the parallelogram with vertices (0, 0), (3, 2), (2, 3), and (-1, 1).

Question 39

Use determinants to find the area of the parallelogram with vertices (4, 1), (6, -1), (4, 0), and (6, 0).

Question 40

Find the area of T (D) where

, D is the rectangle with vertices (0, 0), (4, 0), (4, -1), and (0, -1), and

Question 41

Find the area of T (D) where

, D is the parallelogram with vertices (4, 2), (5, 3), (4, 6), and (3, 5), and

Question 42

For all
matrices A,

.

Question 43

For all
matrices A,

.

Question 44

For all
matrices A,

.

Question 45

If

, then

.

Question 46

.

Question 47

If A is an invertible matrix, then
is invertible, and

.

Question 48

If A is a square matrix and

, then A is a diagonal matrix.