# Chapter 8 - Section 8.5 - Determinants and Cramer's Rule - Exercise Set - Page 946: 41

The determinant is $-42$.

#### Work Step by Step

In order to calculate the given determinant, first evaluate the individual determinant separately. So consider, \begin{align} & \left| \begin{matrix} 3 & 1 \\ -2 & 3 \\ \end{matrix} \right|=3\left( 3 \right)-\left( -2 \right)1 \\ & =9+2 \\ & =11 \end{align} Next consider, \begin{align} & \left| \begin{matrix} 7 & 0 \\ 1 & 5 \\ \end{matrix} \right|=7\left( 5 \right)-1\left( 0 \right) \\ & =35-0 \\ & =35 \end{align} Now find, \begin{align} & \left| \begin{matrix} 3 & 0 \\ 0 & 7 \\ \end{matrix} \right|=3\left( 7 \right)-0 \\ & =21-0 \\ & =21 \end{align} And finally find, \begin{align} & \left| \begin{matrix} 9 & -6 \\ 3 & 5 \\ \end{matrix} \right|=9\left( 5 \right)-3\left( -6 \right) \\ & =45+18 \\ & =63 \end{align} Hence, $\left| \begin{matrix} \left| \begin{matrix} 3 & 1 \\ -2 & 3 \\ \end{matrix} \right| & \left| \begin{matrix} 7 & 0 \\ 1 & 5 \\ \end{matrix} \right| \\ \left| \begin{matrix} 3 & 0 \\ 0 & 7 \\ \end{matrix} \right| & \left| \begin{matrix} 9 & -6 \\ 3 & 5 \\ \end{matrix} \right| \\ \end{matrix} \right|=\left| \begin{matrix} 11 & 35 \\ 21 & 63 \\ \end{matrix} \right|$ Now calculate, \begin{align} & \left| \begin{matrix} 11 & 35 \\ 21 & 63 \\ \end{matrix} \right|=11\left( 63 \right)-21\left( 35 \right) \\ & =693-735 \\ & =-42 \end{align}

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