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Determinant
Math 112, week 7
Goals:
• Cramer’s rule.
• A formula for the inverse of a matrix.
• Determinant as area or volume.
Suggested Textbook Readings: Sections §3.3
Week 7: Determinant
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Cramer’s rule
Let A be an invertible n×n matrix. Then the unique solution ~x of A~x = ~b
is given by
det Ai (~b)
,
det A
i
~
· · · b · · · ~an
xi =
h
~
where Ai (b) = ~a1
i = 1, 2, · · · , n
Example 1: Solve the system by Cramer’s rule.


 2x1 + x2 + x3 = 0
x1 − x2 + 5x3 = 0


x2 − x3 = 4
Week 7: Determinant
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Example 2: Suppose s is an unspecified parameter. Determine the values
of s for which the system has a unique solution, and use Cramer’s rule to
describe
the solution.
(
3sx1 − 2x2 = 4
−6x1 + sx2 = 1
Proof of Cramer’s rule:
Week 7: Determinant
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A formula for A−1 : Let A be an invertible matrix, then
A−1 =
1
adjA
det A

adjoint of A:

 C11 C21 · · · Cn1 


 C12 C22 · · · Cn2 

adjA = 
 ..
..
.. 
 .
.
. 


C1n C2n · · · Cnn
Proof. AA−1 = I. The j-th column of A−1 is a vector ~x that satisfies
A~x = ~ej
Week 7: Determinant
Example 3: Use the formula to find the inverse of


2 1 3 


A = 1 −1 1 


1 4 −2
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Week 7: Determinant
Example
4: A linear system is given below.


 x1 + 2x2 + x3 = 1
2x2 − x3 = −2


−x1 − x2 + x3 = −1
(a) What is the determinant of the coefficient matrix A?
(b) Find the cofactor matrix of A.
(c) Find the inverse of A.
(d) Use the inverse matrix found in (c) to solve the system.
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Week 7: Determinant
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Determinant as area or volume:
• If A is a 2 × 2 matrix, the area of the parallelogram determined by
the columns of A is | det A|.
• If A is a 3 × 3 matrix, the volume of the parallelepiped determined
by the columns of A is | det A|.
Week 7: Determinant
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Example 5: (1) Find the area of the parallelogram whose vertices are
(0, 0), (5, 2), (6, 4), (11, 6).
(2) Find the volume of the parallelepiped with vertices
(0, 0, 0), (1, 0, −3), (1, 2, 4), (5, 1, 0)
(3) Find a formula for the area of the triangle whose vertices are the
origin, ~v1 , ~v2 in R2 .
Week 7: Determinant
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Linear transformation and area, volume. Suppose S is a region in
R2 with finite area or a region in R3 with finite volume.
• Let T : R2 → R2 be a linear transformation with standard matrix A,
then
area of T (S) = | det A| · area of S
• Let T : R3 → R3 be a linear transformation with standard matrix A,
then
volume of T (S) = | det A| · volume of S
Example 6: Find the area of the region E bounded by ellipse whose equation is
x21 x22
+
=1
a2 b2
Week 7: Determinant
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Suppose S is a parallelogram in R2 . Let T : R2 → R2 be a linear transformation with standard matrix A, then
area of T (S) = | det A| · area of S
Proof.
Suppose S is a region in R2 with finite area.