2.1. MATRICES 41
is it possible to multiply these matrices. According to the above discussion it should be a2× 3 matrix of the form
First column︷ ︸︸ ︷(1 2 1
0 2 1
) 1
0
−2
,Second column︷ ︸︸ ︷(
1 2 1
0 2 1
) 2
3
1
,Third column︷ ︸︸ ︷(
1 2 1
0 2 1
) 0
1
1
You know how to multiply a matrix times a vector and so you do so to obtain each of thethree columns. Thus(
1 2 1
0 2 1
) 1 2 0
0 3 1
−2 1 1
=
(−1 9 3
−2 7 3
).
Here is another example.
Example 2.1.7 Multiply the following. 1 2 0
0 3 1
−2 1 1
( 1 2 1
0 2 1
)
First check if it is possible. This is of the form (3× 3) (2× 3) . The inside numbers do notmatch and so you can’t do this multiplication. This means that anything you write will beabsolute nonsense because it is impossible to multiply these matrices in this order. Aren’tthey the same two matrices considered in the previous example? Yes they are. It is justthat here they are in a different order. This shows something you must always rememberabout matrix multiplication.
Order Matters!
Matrix multiplication is not commutative. This is very different than multiplication ofnumbers!
2.1.1 The ijth Entry of a Product
It is important to describe matrix multiplication in terms of entries of the matrices. Whatis the ijth entry of AB? It would be the ith entry of the jth column of AB. Thus it wouldbe the ith entry of Abj . Now
bj =
B1j
...
Bnj
and from the above definition, the ith entry is
n∑k=1
AikBkj . (2.11)