Suppose
A
\displaystyle {A}
A
is a
n
×
n
\displaystyle {n}\times{n}
n
×
n
coefficient matrix. Find the determinants of a matrix that perform the indicated row operations.
R
2
↔
R
3
\displaystyle {R}_{{{2}}}\leftrightarrow{R}_{{{3}}}
R
2
↔
R
3
A
\displaystyle {A}
A
———›
R
A
\displaystyle {R}{A}
R
A
det
R
=
\displaystyle {\det{{R}}}=
det
R
=
−
2
R
2
→
R
2
\displaystyle -{2}{R}_{{{2}}}\rightarrow{R}_{{{2}}}
−
2
R
2
→
R
2
A
\displaystyle {A}
A
———›
S
A
\displaystyle {S}{A}
S
A
det
S
=
\displaystyle {\det{{S}}}=
det
S
=
−
2
R
2
+
R
3
→
R
3
\displaystyle -{2}{R}_{{{2}}}+{R}_{{{3}}}\to{R}_{{{3}}}
−
2
R
2
+
R
3
→
R
3
A
\displaystyle {A}
A
————————›
T
A
\displaystyle {T}{A}
T
A
det
T
=
\displaystyle {\det{{T}}}=
det
T
=
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