8.E: Exercises for Chapter 8

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- 8.2: Determinants
- 9: Inner product spaces

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Calculational Exercises

1. Let $A \in \mathbb{C}^{3\times3}$ be given by

$A = \left[ \begin{array}{ccc} 1 & 0 & i \\ 0 & 1 & 0 \\ -i & 0 & -1 \end{array} \right]$

(a) Calculate $det(A).$
(b) Find $det(A^4 ).$

2. (a) For each permutation $\pi \in \cal{S}_3$ , 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 $3\times3$ matrix.

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 $4\times4$
matrix.

4. Solve for the variable $x$ in the following expression:

$det \left( \left[ \begin{array}{cc} x & -1 \\ 3 & 1-x \end{array} \right] \right) = det \left( \left[ \begin{array}{ccc} 1 & 0 & -3 \\ 2 & x & -6 \\ 1 & 3 & x-5 \end{array} \right] \right).$

5. Prove that the following determinant does not depend upon the value of $\theta$ :

$det \left( \left[ \begin{array}{ccc} sin(\theta) & cos(\theta) & 0 \\ -cos(\theta) & sin(\theta) & 0 \\ sin(\theta) - cos(\theta) & sin(\theta) + cos(\theta) & 1 \end{array} \right] \right)$

6. Given scalars $\alpha, \beta, \gamma \in \mathbb{F}$ , prove that the following matrix is not invertible:

$\left[ \begin{array}{ccc} sin^2 (\alpha) & sin^2 (\beta) & sin^2 (\gamma) \\ cos^2 (\alpha) & cos^2 (\beta) & cos^2 (\gamma) \\ 1 & 1 & 1 \end{array} \right]$

Hint: Compute the determinant.

Proof-Writing Exercises

1. Let $a, b, c, d, e, f \in \mathbb{F}$ be scalars, and suppose that $A$ and $B$ are the following matrices:

$A= \left[ \begin{array}{cc} a & b \\ 0 & c \end{array} \right] ~ \rm{and} ~ B = \left[ \begin{array}{cc} d & e \\ 0 & f \end{array} \right]$

Prove that $AB = BA$ if and only if $det \left( \left[ \begin{array}{cc} b & a-c \\ e & d-f \end{array} \right] \right) = 0.$

2. Given a square matrix $A,$ prove that $A$ is invertible if and only if $A^T A$ is invertible.

3. Prove or give a counterexample: For any $n \geq 1$ and $A, B \in \mathbb(R)^{n \times n}$ , one has

$det(A + B) = det(A) + det(B).$

4. Prove or give a counterexample: For any $r \in \mathbb{R}, n \geq 1$ and $A \in \mathbb{R}^{n \times n} ,$ one has

$det(rA) = r det(A).$

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