11.4.1: Determinants and Cramer's Rule for n x n Matrices (Exercises)
- Page ID
- 108898
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. Let \(A=\left[\begin{array}{ccc}1&2&4\\0&1&3\\-2&5&1\end{array}\right]\). Find the following.
- \(minor(A)_{11}\)
- \(minor(A)_{21}\)
- \(minor(A)_{32}\)
- \(C_{11}\)
- \(C_{21}\)
- \(C_{32}\)
2. Find the determinants of the following matrices.
- \(\left[\begin{array}{ccc}1&2&3\\3&2&2\\0&9&8\end{array}\right]\)
- \(\left[\begin{array}{ccc}4&3&2\\1&7&8\\3&-9&3\end{array}\right]\)
- \(\left[\begin{array}{cccc}1&2&3&2\\1&3&2&3\\4&1&5&0\\1&2&1&2\end{array}\right]\)
- Answer
-
- The answer is \(31\).
- The answer is \(375\).
- The answer is \(-2\).
3. 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
-
\[\left|\begin{array}{ccc}1&2&1\\2&1&3\\2&1&1\end{array}\right|=6\nonumber\]
4. 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
-
\[\left|\begin{array}{ccc}1&2&1\\1&0&1\\2&1&1\end{array}\right|=2\nonumber\]
5. 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
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\[\left|\begin{array}{ccc}1&2&1\\2&1&3\\2&1&1\end{array}\right|=6\nonumber\]
6. 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
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\[\left|\begin{array}{cccc}1&0&0&1\\2&1&1&0\\0&0&0&2\\2&1&3&1\end{array}\right|=-4\nonumber\]
7. Find the determinant of the following matrices.
- \(A=\left[\begin{array}{cc}1&-34\\0&2\end{array}\right]\)
- \(A=\left[\begin{array}{ccc}4&3&14\\0&-2&0\\0&0&5\end{array}\right]\)
- \(A=\left[\begin{array}{cccc}2&3&15&0\\0&4&1&7\\0&0&-3&5\\0&0&0&1\end{array}\right]\)
8. Use Cramer’s rule to find the solution to \[\begin{array}{c}x_1+2x_2+x_3=1 \\ 2x_1-x_2-x_3=2 \\ x_1+x_3=1\end{array}\nonumber\]
- Answer
-
Solution is: \([x_1 = 1, x_2 = 0,x_3 = 0]\). For example, \[y=\frac{\left|\begin{array}{ccc}1&1&1\\2&2&-1\\1&1&1\end{array}\right|}{\left|\begin{array}{ccc}1&2&1\\2&-1&-1\\1&0&1\end{array}\right|}=0\nonumber\]
9. Use Cramer’s rule to find the solution to \[\begin{array}{c}x_1+x_2-2x_3=14 \\ 2x_1-x_2+x_3=0 \\ 6x_1+3x_2+4x_3=1\end{array}\nonumber\]
10. Use Cramer’s rule to find the solution to \[\begin{array}{c}2x_1+x_2+x_3=4 \\ 10x_1-2x_2+2x_3=-1 \\ 6x_1-2x_2+4x_3=8\end{array}\nonumber\]