Suppose A\displaystyle {A} is a n×n\displaystyle {n}\times{n} coefficient matrix. Find the determinants of a matrix that perform the indicated row operations.

R2R3\displaystyle {R}_{{{2}}}\leftrightarrow{R}_{{{3}}}
A\displaystyle {A}———›RA\displaystyle {R}{A}


detR=\displaystyle {\det{{R}}}=

2R2R2\displaystyle -{2}{R}_{{{2}}}\rightarrow{R}_{{{2}}}
A\displaystyle {A}———›SA\displaystyle {S}{A}


detS=\displaystyle {\det{{S}}}=

2R2+R3R3\displaystyle -{2}{R}_{{{2}}}+{R}_{{{3}}}\to{R}_{{{3}}}
A\displaystyle {A}————————›TA\displaystyle {T}{A}
detT=\displaystyle {\det{{T}}}=