Linear Algebra, 2020a

Section IV. Matrix Operations 259
4.9 Example This one happens to start with a row swap.
⎛
⎜
0 3 −1 1 0 0
⎞ ⎛
⎟
⎝1 0 1 0 1 0⎠ ρ 1↔ρ 2 ⎜
1 0 1 0 1 0
⎞
⎟
−→ ⎝0 3 −1 1 0 0⎠
1 −1 0 0 0 1
1 −1 0 0 0 1
⎛
−ρ 1 +ρ 3 ⎜
1 0 1 0 1 0
⎞
⎟
−→ ⎝0 3 −1 1 0 0⎠
0 −1 −1 0 −1 1
.
−→
⎛
⎜
1 0 0 1/4 1/4 3/4
⎞
⎟
⎝0 1 0 1/4 1/4 −1/4⎠
0 0 1 −1/4 3/4 −3/4
4.10 Example This algorithm detects a non-invertible matrix when the left half
won’t reduce to the identity.
(
)
(
1 1 1 0 −2ρ 1 +ρ 2 1 1 1 0
−→
2 2 0 1
0 0 −2 1
)
With this procedure we can give a formula for the inverse of a general 2×2
matrix, which is worth memorizing.
4.11 Corollary The inverse for a 2×2 matrix exists and equals
( ) −1 ( )
a b 1 d −b
=
c d ad − bc −c a
if and only if ad − bc ≠ 0.
Proof This computation is Exercise 21.
QED
We have seen in this subsection, as in the subsection on Mechanics of Matrix
Multiplication, how to exploit the correspondence between linear maps and
matrices. We can fruitfully study both maps and matrices, translating back and
forth to use whichever is handiest.
Over the course of this entire section we have developed an algebra system
for matrices. We can compare it with the familiar algebra of real numbers.
Matrix addition and subtraction work in much the same way as the real number
operations except that they only combine same-sized matrices. Scalar multiplication
is in some ways an extension of real number multiplication. We also have
a matrix multiplication operation and its inverse that are somewhat like the
familiar real number operations (associativity, and distributivity over addition,