"Elementary Transformation"

A = {{r, s, t}, {u, v, w}, {x, y, z}};
Print["A=", MatrixForm[A]];
"Interchage row i, by row j";
"Set i"; i = 2;
"Set j"; j = 1;

"2 steps are involved: 
First, replace row j in A by row i in A to form A1";
A1 = ReplacePart[A, A[[i]], {j}];

      
"Second, replace row i in A1 by row j in A";
A2 = ReplacePart[A1, A[[j]], {i}];
\(\(Print["Exchange", \ " row "\ , \ i, " in A by row ", \ 
        j, \*"": A\!\(\( → \&\(\(R\_j\%i\)\(\\ \\ \\ \)\)\)\)A2 ""];\)\)
Print["A2=", MatrixForm[A2]];

\(\(\(
\)\(\*""Elementary transformation of (Subscript[R, i])(k)""

    \("This multiply row i by a scalar k";\)

    i = 2; k = 4;

    \(Print["i=", i, " ,k=", k];\)

    \(Print[MatrixForm[ReplacePart[A, k\ A[\([i]\)], {i}]]];\)\)\)\)

"Elementary transformation of \!\(R\_i\)(k)"
"A="(({{r, s, t}, {u, v, w}, {x, y, z}}))

\(\*""Elementary transformation of \!\(R\_j\%i\)(k)""\)
"This add to row j by k*(row i)";
j = 3; i = 2; k = 6;

Print["i=", i, "; j=", j,  " ;k=", 
        k];
Print[
        MatrixForm[
          ReplacePart[A, 
            A[[j]] + k A[[i]], {j}]]];

"A="(({{r, s, t}, {u, v, w}, {x, y, z}}))

\(\(\(\*""If A (Overscript[→ , (Subscript[R, j]^i) (k)])B,|A|=|B|,pg 15 (ix)"";\)\(
\)
    \)\)
Print["Det[A]=", Det[A]];
B = 
        ReplacePart[A, 
          A[[j]] + k A[[i]], {j}];
\(\(Print[\*""A (Overscript[→ , (Subscript[R, j]^i) (k)])B="" MatrixForm[B]];\)\)
Print["Det[B]=", Det[B]];
\(\(Print[\*""This examplifies Det[A]= Det[B], where A (Overscript[→ , (Subscript[R, j]^i) (k)])B""];\)\)