Skip to main content
Mathematics LibreTexts

8.E: Exercises for Chapter 8

  • Page ID
    287
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    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

    \[det(rA) = r det(A).\]


    This page titled 8.E: Exercises for Chapter 8 is shared under a not declared license and was authored, remixed, and/or curated by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling.

    • Was this article helpful?