Lesson test: Determinants and inverses

1.[1p]
What is the determinant of $[(4, 2), (1, 3)]$ ?

CorrectNot quite: 10

2.[2p]
Geometrically, the determinant of a two by two matrix is

the signed area of the parallelogram the unit square becomes

Correct

the length of the longer column

The answer is: the signed area of the parallelogram the unit square becomes

the angle between the two columns

The answer is: the signed area of the parallelogram the unit square becomes

the sum of the diagonal entries

The answer is: the signed area of the parallelogram the unit square becomes

3.[2p]
A matrix with determinant zero has dependent columns and cannot be inverted.

True

Correct

False

The answer is: True

4.[2p]
If $\det A = 5$ and $\det B = 8$ , what is $\det (A B)$ ?

CorrectNot quite: 40

5.[2p]
What does a negative determinant tell you?

The transformation reverses orientation, flipping the plane over

Correct

The transformation shrinks every area

The answer is: The transformation reverses orientation, flipping the plane over

The matrix has no inverse

The answer is: The transformation reverses orientation, flipping the plane over

The columns are perpendicular

The answer is: The transformation reverses orientation, flipping the plane over

6.[3p]
What is the determinant of the matrix with rows $(1, 2, 3)$ , $(0, 1, 4)$ and $(5, 6, 0)$ ?

CorrectNot quite: 1

7.[3p]
The inverse of $[(2, 1), (5, 3)]$ is $[(3, - 1), (b, 2)]$ . What is $b$ ?

CorrectNot quite: -5

8.[3p]
Which of these are equivalent to a square matrix being invertible?

Select all that apply

Its determinant is non-zero

Correct

Its columns are linearly independent

Correct

$A x = 0$ has only the solution $x = 0$

Correct

Its entries are all non-zero

The answer is: Its determinant is non-zero, Its columns are linearly independent, $A\mathbf{x} = \mathbf{0}$ has only the solution $\mathbf{x} = \mathbf{0}$

9.[3p]
Why is a small determinant a poor test for near-singularity in floating point?

Scaling a well-behaved matrix by $0.1$ multiplies its determinant by ${0.1}^{n}$ , which underflows without any loss of invertibility

Correct

Determinants of large matrices are always negative

The answer is: Scaling a well-behaved matrix by $0.1$ multiplies its determinant by $0.1^n$, which underflows without any loss of invertibility

The determinant cannot be computed for matrices larger than three by three

The answer is: Scaling a well-behaved matrix by $0.1$ multiplies its determinant by $0.1^n$, which underflows without any loss of invertibility

Rounding makes every determinant exactly zero

The answer is: Scaling a well-behaved matrix by $0.1$ multiplies its determinant by $0.1^n$, which underflows without any loss of invertibility

1.[1p] What is the determinant of [(4,2),(1,3)]?

2.[2p] Geometrically, the determinant of a two by two matrix is

3.[2p] A matrix with determinant zero has dependent columns and cannot be inverted.

4.[2p] If detA=5 and detB=8, what is det(AB)?

5.[2p] What does a negative determinant tell you?

6.[3p] What is the determinant of the matrix with rows (1,2,3), (0,1,4) and (5,6,0)?

7.[3p] The inverse of [(2,1),(5,3)] is [(3,-1),(b,2)]. What is b?

8.[3p] Which of these are equivalent to a square matrix being invertible?

9.[3p] Why is a small determinant a poor test for near-singularity in floating point?

1.[1p]
What is the determinant of $[(4, 2), (1, 3)]$ ?

2.[2p]
Geometrically, the determinant of a two by two matrix is

3.[2p]
A matrix with determinant zero has dependent columns and cannot be inverted.

4.[2p]
If $\det A = 5$ and $\det B = 8$ , what is $\det (A B)$ ?

5.[2p]
What does a negative determinant tell you?

6.[3p]
What is the determinant of the matrix with rows $(1, 2, 3)$ , $(0, 1, 4)$ and $(5, 6, 0)$ ?

7.[3p]
The inverse of $[(2, 1), (5, 3)]$ is $[(3, - 1), (b, 2)]$ . What is $b$ ?

8.[3p]
Which of these are equivalent to a square matrix being invertible?

9.[3p]
Why is a small determinant a poor test for near-singularity in floating point?