6.3 Cramer’s Rule
and Geometric
Interpretations
of a Determinant
Finding area
• The determinant of a 2x2 matrix
can be interpreted as the area of a
Parallelogram
(note the absolute values of the
determinant gives the indicated area)
• find the area of a parallelogram
• (see next slide for explanation)
For more information visit
http://www-math.mit.edu/18.013A/HTML/chapter04/section01.html
• 2 × 2 determinants and area
• Recall that the area of the parallelogram spanned by a and b
is the magnitude of a×b. We can write the cross product of a
= a1i + a2j + a3k and b = b1i + b2j + b3k as the determinant
• a×b= .
• Now, imagine that a and b lie in the plane so that
a3 = b3 = 0. Using our rules for calculating determinants we
see that, in this case, the cross product simplifies to
• a × b = k.
• Hence, the area of the parallelogram, ||a × b||, is the absolute
value of the determinant
Volume
Determinants can also be used to find the
volume of a parallelepiped
Given the following matrix:
Det(A) is can be interpreted as the
volume of the parallelpiped shown at
the right.
• 3 × 3 determinants and volume
• The volume of a parallelepiped spanned by the vectors
a, b and c is the absolute value of the scalar triple
product (a × b) ⋅ c. We can write the scalar triple product
of a = a1i + a2j + a3k,
b = b1i + b2j + b3k, and c = c1i + c2j + c3k as the
determinant
• (a × b) ⋅ c = .Hence, the volume of the parallelepiped
spanned by a, b, and c is |(a × b) ⋅ c| = .
How do determinants expand into
higher dimensions?
We can not fully prove this until after chapter
(a proof is on p. 276 of the text) However if
the determinant of a matrix is zero then the
vectors do not fill the entire region.
(analogous to zero area or zero volume)
Cramer’s Rule
If one solves this system using augmented
matrices the solution to this system is
Provided that
Another way to find the solution is with
determinants
Cramer’s Rule states that
and
Where D is the determinant of A
Dx is the Determinant of A with the x column replaced by b
Dy is the Determinant of A with the y column replaced by b
Note: verify that this works by checking with the previous slide
Homework:
p. 607 (8.5) Pre-Calc book 1-27 odd
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