Evaluate B left begin matrix 2 amp 3 amp - 4 1 ... - SS3 Mathematics Matrices and Determinants Question

Using the third row, \(|A| = a_{31}C_{31} + a_{32}C_{32} + a_{33}C_{33}\)

\[C_{31} = {( - 1)}^{3 + 1}\left| \begin{matrix} 3 & - 4 \\ 2 & 3 \\ \end{matrix} \right| = ( - 1)^{4}\left| \begin{matrix} 3 & - 4 \\ 2 & 3 \\ \end{matrix} \right| = \ + \left\lbrack 3(3) - ( - 4)2 \right\rbrack = + \lbrack 9 + 8\rbrack = 17\]

\[C_{12} = {( - 1)}^{3 + 2}\left| \begin{matrix} 2 & - 4 \\ 1 & 3 \\ \end{matrix} \right| = ( - 1)^{5}\left| \begin{matrix} 2 & - 4 \\ 1 & 3 \\ \end{matrix} \right| = \ - \left\lbrack 2(3) - ( - 4)(1) \right\rbrack = - \lbrack 6 + 4\rbrack = - 10\]

\[C_{13} = {( - 1)}^{3 + 3}\left| \begin{matrix} 2 & 3 \\ 1 & 2 \\ \end{matrix} \right| = ( - 1)^{6}\left| \begin{matrix} 2 & 3 \\ 1 & 2 \\ \end{matrix} \right| = \ + \left\lbrack 2(2) - 3(1) \right\rbrack = + \lbrack 4 - 3\rbrack = 1\]

\[|A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} = 3(17) + ( - 1)( - 10) + ( - 1)1 = \ 51 + 10 - 1 = 60\]