Linear Algebra, 2020a

Section IV. Matrix Operations 245
The striking thing about this operation is the way that rows and columns
combine. The i, j entry of the matrix product is the dot product of row i of the
left matrix with column j of the right one. For instance, here a second row and
a third column combine to make a 2, 3 entry.
⎛ ⎞
1 1
⎜ ⎟
⎝0 1⎠
1 0
(
4
5
6
7
8
9
)
2
=
3
⎛
⎜
9 13 17 5
⎞
⎟
⎝5 7 9 3⎠
4 6 8 2
We can view this as the left matrix acting by multiplying its rows into the
columns of the right matrix. Or, it is the right matrix using its columns to act
on the rows of the left matrix. Below, we will examine actions from the left and
from the right for some simple matrices.
Simplest is the zero matrix.
3.1 Example Multiplying by a zero matrix from the left or from the right results
in a zero matrix.
(
0 0
0 0
)(
1 3 2
−1 1 −1
) (
=
0 0 0
0 0 0
) (
2 3
1 4
)(
0 0
0 0
) (
=
The next easiest matrices are the ones with a single nonzero entry.
0 0
0 0
3.2 Definition A matrix with all 0’s except for a 1 in the i, j entry is an i, j unit
matrix (or matrix unit).
3.3 Example This is the 1, 2 unit matrix with three rows and two columns,
multiplying from the left.
⎛ ⎞
⎛
0 1 ( )
⎜ ⎟ 5 6 ⎜
7 8
⎞
⎟
⎝ 0 0⎠
= ⎝0 0⎠
7 8
0 0
0 0
Acting from the left, an i, j unit matrix copies row j of the multiplicand into
row i of the result. From the right an i, j unit matrix picks out column i of the
multiplicand and copies it into column j of the result.
⎛
⎜
1 2 3
⎞ ⎛ ⎞ ⎛
0 1
⎟ ⎜ ⎟ ⎜
0 1
⎞
⎟
⎝4 5 6⎠
⎝0 0⎠ = ⎝0 4⎠
7 8 9 0 0 0 7
3.4 Example Rescaling unit matrices simply rescales the result. This is the action
from the left of the matrix that is twice the one in the prior example.
⎛
⎜
0 2
⎞
⎛ ⎞
( ) 14 16
⎟ 5 6 ⎜ ⎟
⎝0 0⎠
= ⎝ 0 0 ⎠
7 8
0 0
0 0
)