Skip to main content
Mathematics LibreTexts

3.1E: Exercises for Section 3.1

  • Page ID
    197414
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\dsum}{\displaystyle\sum\limits} \)

    \( \newcommand{\dint}{\displaystyle\int\limits} \)

    \( \newcommand{\dlim}{\displaystyle\lim\limits} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \(\newcommand{\longvect}{\overrightarrow}\)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
    Exercise \(\PageIndex{1}\)

    Find the determinants of the following matrices.

    1. \(\left[\begin{array}{cc}1&3\\0&2\end{array}\right]\)
    2. \(\left[\begin{array}{cc}0&3\\0&2\end{array}\right]\)
    3. \(\left[\begin{array}{cc}4&3\\6&2\end{array}\right]\)
    Exercise \(\PageIndex{2}\)

    Let \(A=\left[\begin{array}{ccc}1&2&4\\0&1&3\\-2&5&1\end{array}\right]\). Find the following.

    1. \(\mathrm{minor}(A)_{11}\)
    2. \(\mathrm{minor}(A)_{21}\)
    3. \(\mathrm{minor}(A)_{32}\)
    4. \(\mathrm{cof}(A)_{11}\)
    5. \(\mathrm{cof}(A)_{21}\)
    6. \(\mathrm{cof}(A)_{32}\)
    Exercise \(\PageIndex{3}\)

    Find the determinants of the following matrices.

    1. \(\left[\begin{array}{ccc}1&2&3\\3&2&2\\0&9&8\end{array}\right]\)
    2. \(\left[\begin{array}{ccc}4&3&2\\1&7&8\\3&-9&3\end{array}\right]\)
    3. \(\left[\begin{array}{cccc}1&2&3&2\\1&3&2&3\\4&1&5&0\\1&2&1&2\end{array}\right]\)
    Answer
    1. The answer is \(31\).
    2. The answer is \(375\).
    3. The answer is \(-2\).
    Exercise \(\PageIndex{4}\)

    Find the following determinant by expanding along the first row and second column. \[\left|\begin{array}{ccc}1&2&1\\2&1&3\\2&1&1\end{array}\right|\nonumber\]

    Answer

    \[\left|\begin{array}{ccc}1&2&1\\2&1&3\\2&1&1\end{array}\right|=6\nonumber\]

    Exercise \(\PageIndex{5}\)

    Find the following determinant by expanding along the first column and third row. \[\left|\begin{array}{ccc}1&2&1\\1&0&1\\2&1&1\end{array}\right|\nonumber\]

    Answer

    \[\left|\begin{array}{ccc}1&2&1\\1&0&1\\2&1&1\end{array}\right|=2\nonumber\]

    Exercise \(\PageIndex{6}\)

    Find the following determinant by expanding along the second row and first column. \[\left|\begin{array}{ccc}1&2&1\\2&1&3\\2&1&1\end{array}\right|\nonumber\]

    Answer

    \[\left|\begin{array}{ccc}1&2&1\\2&1&3\\2&1&1\end{array}\right|=6\nonumber\]

    Exercise \(\PageIndex{7}\)

    Compute the determinant by cofactor expansion. Pick the easiest row or column to use. \[\left|\begin{array}{cccc}1&0&0&1\\2&1&1&0\\0&0&0&2\\2&1&3&1\end{array}\right|\nonumber\]

    Answer

    \[\left|\begin{array}{cccc}1&0&0&1\\2&1&1&0\\0&0&0&2\\2&1&3&1\end{array}\right|=-4\nonumber\]

    Exercise \(\PageIndex{8}\)

    Find the determinant of the following matrices.

    1. \(A=\left[\begin{array}{cc}1&-34\\0&2\end{array}\right]\)
    2. \(A=\left[\begin{array}{ccc}4&3&14\\0&-2&0\\0&0&5\end{array}\right]\)
    3. \(A=\left[\begin{array}{cccc}2&3&15&0\\0&4&1&7\\0&0&-3&5\\0&0&0&1\end{array}\right]\)
    Exercise \(\PageIndex{9}\)

    Prove that for any \( 2 \times 2 \) or \( 3 \times 3 \) matrix \(A\), if one row is a multiple of another row, then the \( \det (A) =0\)

    Exercise \(\PageIndex{10}\)

    Find the value of \(k\) such that \( \det (A)=0\) for the following matrix.

    \[\left|\begin{array}{ccc}k&4&7\\-2&0&8\\0&-6&9\end{array}\right|\nonumber\]

    Answer

    \(k = -\frac{13}{4}\)

    Exercise \(\PageIndex{11}\)

    Suppose \(A\) is a \( 2 \times 2 \) matrix. What effect will the following have on the determinant of the resulting matrix?

    1. Multiplying the first row by \(2\).
    2. Multiplying the second column by \(-3\).
    3. Multiplying the entire matrix by \(5\).
    Answer
    1. The determinant will be multiplied by a factor of \(2\).
    2. The determinant will be multiplied by a factor of \(-3\).
    3. The determinant will be multiplied by a factor of \(5^2\).
    Exercise \(\PageIndex{12}\)

    Find the value of \(c\) such that \( \det (A) \neq 0\) for the following matrix.

    \[\left|\begin{array}{cccc}3&2&0&6\\-1&5&6&c\\0&-2&-2&0\\4&-3&1&8 \end{array}\right|\nonumber\]

    Answer

    \(c \neq -2\)


    This page titled 3.1E: Exercises for Section 3.1 is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Doli Bambhania, Fatemeh Yarahmadi, and Bill Wilson.

    • Was this article helpful?