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Mathematics LibreTexts

8.E: Exercises for Chapter 8

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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