Question 10.4.3: Evaluating a 3 by 3 Determinant Find the determinant of A = ...

Evaluating a 3 by 3 Determinant

Find the determinant of A=[234122341]A=\left[\begin{array}{rrr}2 & 3 & 4 \\1 & -2 & 2 \\3 & 4 & -1\end{array}\right].

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Expanding by the first row, we have A=a11A11+a12A12+a13A13, |A|=a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13}, where

A11=(1)1+1M11=(1)2241A12=(1)1+2M12=(1)1231A13=(1)1+3M13=(1)1234\begin{aligned}&A_{11}=(-1)^{1+1} M_{11}=(1)\left|\begin{array}{rr}-2 & 2 \\4 & -1\end{array}\right| \\&A_{12}=(-1)^{1+2} M_{12}=(-1)\left|\begin{array}{rr}1 & 2 \\3 & -1\end{array}\right| \\&A_{13}=(-1)^{1+3} M_{13}=(1)\left|\begin{array}{rr}1 & -2 \\3 & 4\end{array}\right|\end{aligned}

Thus,

234122341=2(1)2241+3(1)1231+4(1)1234\left|\begin{array}{rrr}2 & 3 & 4 \\1 & -2 & 2 \\3 & 4 & -1\end{array}\right|=2(1)\left|\begin{array}{rr}-2 & 2 \\4 & -1\end{array}\right|+3(-1)\left|\begin{array}{rr}1 & 2 \\3 & -1\end{array}\right|+4(1)\left|\begin{array}{rr}1 & -2 \\3 & 4\end{array}\right|

=2(28)3(16)+4(4+6)=2(2-8)-3(-1-6)+4(4+6)          Evaluate determinants of order 2.

= -12 + 21 + 40 = 49              Simplify.

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