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
1
↔
R
4
\displaystyle {R}_{{{1}}}\leftrightarrow{R}_{{{4}}}
R
1
↔
R
4
A
\displaystyle {A}
A
———›
R
A
\displaystyle {R}{A}
R
A
det
R
=
\displaystyle {\det{{R}}}=
det
R
=
5
R
1
→
R
1
\displaystyle {5}{R}_{{{1}}}\rightarrow{R}_{{{1}}}
5
R
1
→
R
1
A
\displaystyle {A}
A
———›
S
A
\displaystyle {S}{A}
S
A
det
S
=
\displaystyle {\det{{S}}}=
det
S
=
5
R
1
+
R
4
→
R
4
\displaystyle {5}{R}_{{{1}}}+{R}_{{{4}}}\to{R}_{{{4}}}
5
R
1
+
R
4
→
R
4
A
\displaystyle {A}
A
————————›
T
A
\displaystyle {T}{A}
T
A
det
T
=
\displaystyle {\det{{T}}}=
det
T
=
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