Linear Algebra, 2020a

Section IV. Matrix Operations 249
3.15 Example From the left these matrices permute rows.
⎛
⎜
0 0 1
⎞ ⎛
⎟ ⎜
1 2 3
⎞ ⎛
⎟ ⎜
7 8 9
⎞
⎟
⎝1 0 0⎠
⎝4 5 6⎠ = ⎝1 2 3⎠
0 1 0 7 8 9 4 5 6
From the right they permute columns.
⎛
⎜
1 2 3
⎞ ⎛
⎟ ⎜
0 0 1
⎞ ⎛
⎟ ⎜
2 3 1
⎞
⎟
⎝4 5 6⎠
⎝1 0 0⎠ = ⎝5 6 4⎠
7 8 9 0 1 0 8 9 7
We finish this subsection by applying these observations to get matrices that
perform Gauss’s Method and Gauss-Jordan reduction. We have already seen
how to produce a matrix that rescales rows, and a row swapper.
3.16 Example Multiplying by this matrix rescales the second row by three.
⎛
⎜
1 0 0
⎞ ⎛
⎞ ⎛
0 2 1 1
⎟ ⎜
⎟ ⎜
0 2 1 1
⎞
⎟
⎝0 3 0⎠
⎝0 1/3 1 −1⎠ = ⎝0 1 3 −3⎠
0 0 1 1 0 2 0 1 0 2 0
3.17 Example This multiplication swaps the first and third rows.
⎛
⎜
0 0 1
⎞ ⎛
⎟ ⎜
0 2 1 1
⎞ ⎛
⎟ ⎜
1 0 2 0
⎞
⎟
⎝0 1 0⎠
⎝0 1 3 −3⎠ = ⎝0 1 3 −3⎠
1 0 0 1 0 2 0 0 2 1 1
To see how to perform a row combination, we observe something about those
two examples. The matrix that rescales the second row by a factor of three
arises in this way from the identity.
⎛
⎜
1 0 0
⎞ ⎛
⎟
⎝0 1 0⎠ 3ρ 2 ⎜
1 0 0
⎞
⎟
−→ ⎝0 3 0⎠
0 0 1
0 0 1
Similarly, the matrix that swaps first and third rows arises in this way.
⎛
⎜
1 0 0
⎞ ⎛
⎟
⎝0 1 0⎠ ρ 1↔ρ 3 ⎜
0 0 1
⎞
⎟
−→ ⎝0 1 0⎠
0 0 1
1 0 0
3.18 Example The 3×3 matrix that arises as
⎛
⎜
1 0 0
⎞
⎛
⎟
⎝0 1 0⎠ −2ρ 2+ρ 3 ⎜
1 0 0
⎞
⎟
−→ ⎝0 1 0⎠
0 0 1
0 −2 1