Linear Algebra, 2020a

370 Chapter Four. Determinants
(
The second is defined by the vectors x 1
(
1 3)
and
2
1)
and one of the properties of
the size function — the determinant — is that therefore the size of the second
box is x 1 times the size of the first. The third box is derived from the second by
(
shearing, adding x 2
(
2 1)
to
1
(
x1
3)
to get
1
(
x1
3)
+
2
(
x2
1)
=
6
(
8)
, along with
2
1)
. The
determinant is not affected by shearing so the size of the third box equals that
of the second.
Taken together we have this.
∣ ∣ ∣ x 1 ·
1 2
∣∣∣∣ ∣3 1∣ = x 1 · 1 2
∣∣∣∣ x 1 · 3 1∣ = x 1 · 1 + x 2 · 2 2
∣∣∣∣ x 1 · 3 + x 2 · 1 1∣ = 6 2
8 1∣
Solving gives the value of one of the variables.
6 2
∣8 1∣
x 1 =
1 2
= −10
−5 = 2
∣3 1∣
The generalization of this example is Cramer’s Rule: if |A| ≠ 0 then the
system A⃗x = ⃗b has the unique solution x i = |B i |/|A| where the matrix B i is
formed from A by replacing column i with the vector ⃗b. The proof is Exercise 3.
For instance, to solve this system for x 2
⎛
⎜
1 0 4
⎞ ⎛ ⎞ ⎛ ⎞
x 1 2
⎟ ⎜ ⎟ ⎜ ⎟
⎝2 1 −1⎠
⎝x 2 ⎠ = ⎝ 1 ⎠
1 0 1 x 3 −1
we do this computation.
1 2 4
2 1 −1
∣1 −1 1 ∣
x 2 =
1 0 4
2 1 −1
∣1 0 1 ∣
= −18
−3
Cramer’s Rule lets us by-eye solve systems that are small and simple. For
example, we can solve systems with two equations and two unknowns, or three
equations and three unknowns, where the numbers are small integers. Such
cases appear often enough that many people find this formula handy.
But using it to solving large or complex systems is not practical, either by
hand or by a computer. A Gauss’s Method-based approach is faster.
Exercises
1 Use Cramer’s Rule to solve each for each of the variables.