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.

R1R4\displaystyle {R}_{{{1}}}\leftrightarrow{R}_{{{4}}}
A\displaystyle {A}———›RA\displaystyle {R}{A}


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

5R1R1\displaystyle {5}{R}_{{{1}}}\rightarrow{R}_{{{1}}}
A\displaystyle {A}———›SA\displaystyle {S}{A}


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

5R1+R4R4\displaystyle {5}{R}_{{{1}}}+{R}_{{{4}}}\to{R}_{{{4}}}
A\displaystyle {A}————————›TA\displaystyle {T}{A}
detT=\displaystyle {\det{{T}}}=