So this is my process, but I used calculator to check if I got them right or not. and it seems like i got both of them wrong. number 6 supposed to be 36 and 1 supposed to be 135. can anyone please explain where did i do wrong?
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$\begingroup$The operation $\frac{1}{3}R_3$ changes the determinant, dividing it by $3$; similarly, the operation $R_2+2R_3\to R_3$ multiplies the determinant by $2$.
Further, the last operation leaves $1$ in position $(3,3)$. As a consequence, the determinant is $$ 1\cdot(-2)\cdot1\cdot 3\cdot\frac{1}{2}=-3 $$ which the computer confirms:
GP/PARI CALCULATOR Version 2.11.0 (released) i386 running darwin (x86-64/GMP-6.1.2 kernel) 64-bit version compiled: Jul 19 2018, Apple LLVM version 9.0.0 (clang-900.0.39.2) threading engine: single (readline v7.0 enabled, extended help enabled) Copyright (C) 2000-2018 The PARI Group PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER. Type ? for help, \q to quit. Type ?17 for how to get moral (and possibly technical) support. parisize = 8000000, primelimit = 500000 ? a=[1,-3,-3;-1,1,2;2,-3,-3] %1 = [ 1 -3 -3] [-1 1 2] [ 2 -3 -3] ? matdet(a) %2 = -3 The same program confirms that the second determinant is $135$; indeed, the last operation you do multiplies the determinant by $-29$; the others don't modify it.
$\endgroup$1$\begingroup$For the first determinant, there is a final error: the last determinant should be $$\begin{vmatrix} 1&-3&-3 \\ 0&-2&-1 \\ 0&\phantom{-}0&\phantom{-}\color{red}1 \end{vmatrix}, $$ so the final determinant is $-2$.
There are also two conceptual errors:
- The step $\frac13 R_3\to R_3$ multiplies the determinant by $\frac13$;
- the step $R_2+2R_3\to R_3$ multiplies the determinant by $2$.
So to get the original determinant, you have to multiply the final determinant by $\frac32$, which yields $\color{red}{-3}$ as the sought determinant.
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