Page 397 - 35Linear Algebra

G.7 Determinants 397

2. As a matrix equation
MX = V ,
which we would solve by finding M −1 (again, if it exists), so that

X = M −1 V .

3. As a linear transformation
n n
L : R −→ R
via
n
n
R 3 X 7−→ MX ∈ R .
n
In this case we have to study the equation L(X) = V because V ∈ R .
Lets focus on the first two methods. In particular we want to think about
how the augmented matrix method can give information about finding M −1 . In
particular, how it can be used for handling determinants.
The main idea is that the row operations changed the augmented matrices,
but we also know how to change a matrix M by multiplying it by some other
matrix E, so that M → EM. In particular can we find ‘‘elementary matrices’’
the perform row operations?
Once we find these elementary matrices is is very important to ask how they
effect the determinant, but you can think about that for your own self right
now.
Lets tabulate our names for the matrices that perform the various row
operations:

Row operation Elementary Matrix

R i ↔ R j E j i
i
R i → λR i R (λ)
i
S (λ)
R i → R i + λR j
j
To finish off the video, here is how all these elementary matrices work
for a 2 × 2 example. Lets take

a b
M = .
c d
A good thing to think about is what happens to det M = ad − bc under the
operations below.
• Row swap:

0 1 1 0 1 a b c d
1
E = , E M = = .
2 1 0 2 1 0 c d a b
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