Linear Algebra

Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling

Chapter 8 Permutations and the Determinant - all with Video Answers

Educators

Chapter Questions

01:48

Problem 1

Let $A \in \mathbb{C}^{3 \times 3}$ be given by
$$A=\left[\begin{array}{ccc}1 & 0 & i \\0 & 1 & 0 \\-i & 0 & -1
\end{array}\right]$$
(a) Calculate $\operatorname{det}(A)$.
(b) Find $\operatorname{det}\left(A^{4}\right)$.

Robert Daugherty

Numerade Educator

02:25

Problem 2

(a) For each permutation $\pi \in \mathcal{S}_{3}$, compute the number of inversions in $\pi$, and classify $\pi$ as being either an even or an odd permutation.
(b) Use your result from Part (a) to construct a formula for the determinant of a $3 \times 3$ matrix.

James Kiss

Numerade Educator

03:01

Problem 3

a) For each permutation $\pi \in S_{4}$, compute the number of inversions in $\pi$, and classify $\pi$ as being either an even or an odd permutation.
(b) Use your result from Part (a) to construct a formula for the determinant of a $4 \times 4$ matrix.

Lindsay El

Numerade Educator

04:16

Problem 4

Solve for the variable $x$ in the following expression:
$$\operatorname{det}\left(\left[\begin{array}{cc}x & -1 \\
3 & 1-x\end{array}\right]\right)=\operatorname{det}\left(\left[\begin{array}{ccc}1 & 0 & -3 \\2 & x & -6 \\1 & 3 & x-5\end{array}\right]\right)$$

Thomas Emment

Numerade Educator

01:04

Problem 5

Prove that the following determinant does not depend upon the value of $\theta$ :
$$\operatorname{det}\left(\left[\begin{array}{ccc}
\sin (\theta) & \cos (\theta) & 0 \\
-\cos (\theta) & \sin (\theta) & 0 \\\sin (\theta)-\cos (\theta) & \sin (\theta)+\cos (\theta) & 1\end{array}\right]\right)
$$

Tanishq Gupta

Numerade Educator

02:57

Problem 6

Given scalars $\alpha, \beta, \gamma \in \mathbb{F}$, prove that the following matrix is not invertible:
$$\left[\begin{array}{ccc}\sin ^{2}(\alpha) & \sin ^{2}(\beta) & \sin ^{2}(\gamma) \\
\cos ^{2}(\alpha) & \cos ^{2}(\beta) & \cos ^{2}(\gamma) \\1 & 1 & 1
\end{array}\right]$$

Lucía Guerrero

Numerade Educator