- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
3.6. $\quad$ Exercises
$\quad$ In this set of exercises you may assume existence of a determinant function. Determinants of order three may be computed by Equation (3.2).
\begin{align*}
(3.2) \qquad
\det\begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
&=
a_{11}\begin{vmatrix}
a_{22} & a_{23} \\ a_{32} & a_{33}
\end{vmatrix}
-
a_{12}\begin{vmatrix}
a_{21} & a_{23} \\ a_{31} & a_{33}
\end{vmatrix}
+
a_{13}\begin{vmatrix}
a_{21} & a_{22} \\ a_{31} & a_{32}
\end{vmatrix}
\end{align*}
1. $\quad$ Compute each of the following determinants. \begin{align*} \text{(a)} \quad \begin{vmatrix} 2 & 1 & 1 \\ 1 & 4 & -4 \\ 1 & 0 & 2 \end{vmatrix}, \quad \text{(b)} \quad \begin{vmatrix} 3 & 0 & 8 \\ 5 & 0 & 7 \\ -1 & 4 & 2 \end{vmatrix}, \quad \text{(c)} \quad \begin{vmatrix} a & 1 & 0 \\ 2 & a & 2 \\ 0 & 1 & a \end{vmatrix}. \end{align*}
Solution. $\quad$
(a) \begin{align*} \begin{vmatrix} 2 & 1 & 1 \\ 1 & 4 & -4 \\ 1 & 0 & 2 \end{vmatrix} &= 2\begin{vmatrix}4 & - 4 \\ 0 & 2\end{vmatrix} - 1\begin{vmatrix}1 & -4 \\ 1 & 2\end{vmatrix} + 1\begin{vmatrix}1 & 4 \\ 1 & 0 \end{vmatrix} \\ &= 2(8 + 0) - (2 + 4) + (0 - 4) \\ &= 6. \end{align*}
(b) \begin{align*} \begin{vmatrix} 3 & 0 & 8 \\ 5 & 0 & 7 \\ -1 & 4 & 2 \end{vmatrix} &= 3\begin{vmatrix} 0 & 7 \\ 4 & 2 \end{vmatrix} - 0\begin{vmatrix} 5 & 7 \\ -1 & 2 \end{vmatrix} + 8\begin{vmatrix} 5 & 0 \\ -1 & 4 \end{vmatrix} \\ &= 3(0 - 28) - 0(10 + 7) + 8(20 - 0) \\ &= 76. \end{align*}
(c) \begin{align*} \begin{vmatrix} a & 1 & 0 \\ 2 & a & 2 \\ 0 & 1 & a \end{vmatrix} &= a\begin{vmatrix} a & 2 \\ 1 & a\end{vmatrix} - \begin{vmatrix} 2 & 2 \\ 0 & a \end{vmatrix} + 0\begin{vmatrix} 2 & a \\ 0 & 1 \end{vmatrix} \\ &= a(a^2 - 2) - (2a - 0) + 0(2 - 0) \\ &= a^3 - 4a. \quad \blacksquare \end{align*}