Linear Algebra

Section IV. Matrix Operations 227
3.19 Lemma Gaussian reduction can be done through matrix multiplication.
(1) If H kρi
−→ G then Mi(k)H = G.
(2) If H ρi↔ρj
−→ G then Pi,jH = G.
(3) If H kρi+ρj
−→ G then Ci,j(k)H = G.
Proof. Clear. QED
3.20 Example This is the first system, from the first chapter, on which we
performed Gauss’ method.
3x3 =9
x1 +5x2 − 2x3 =2
(1/3)x1 +2x2 =3
It can be reduced with matrix multiplication. Swap the first and third rows,
⎛
0
⎝0
0
1
⎞ ⎛
1 0
0⎠⎝1
0
5
3
−2
⎞ ⎛
9 1/3
2⎠
= ⎝ 1
2
5
0
−2
⎞
3
2⎠
1 0 0 1/3 2 0 3 0 0 3 9
triple the first row,
⎛
3
⎝0
0
1
⎞ ⎛
0 1/3
0⎠⎝1
2
5
0
−2
⎞ ⎛
3 1
2⎠
= ⎝1
6
5
0
−2
⎞
9
2⎠
0 0 1 0 0 3 9 0 0 3 9
and then add −1 times the first row to the second.
⎛
1
⎝−1
0
1
⎞ ⎛
0 1
0⎠⎝1
6
5
0
−2
⎞ ⎛
9 1
2⎠
= ⎝0
6
−1
0
−2
⎞
9
−7⎠
0 0 1 0 0 3 9 0 0 3 9
Now back substitution will give the solution.
3.21 Example Gauss-Jordan reduction works the same way. For the matrix
ending the prior example, first adjust the leading entries
⎛
1
⎝0 0
−1
⎞ ⎛
0 1
0 ⎠ ⎝0 6
−1
0
−2
⎞ ⎛
9 1
−7⎠
= ⎝0 6
1
0
2
⎞
9
7⎠
0 0 1/3 0 0 3 9 0 0 1 3
and to finish, clear the third column and then the second column.
⎛
1
⎝0 −6
1
⎞ ⎛
0 1
0⎠⎝0
0
1
⎞ ⎛
0 1
−2⎠
⎝0 6
1
0
2
⎞ ⎛
9 1
7⎠
= ⎝0 0
1
0
0
⎞
3
1⎠
0 0 1 0 0 1 0 0 1 3 0 0 1 3