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6.3 Cramer

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Title: 6.3 Cramer s Rule and Geometric Interpretations of a Determinant Last modified by: neilw Document presentation format: On-screen Show Company – PowerPoint PPT presentation

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Title: 6.3 Cramer


1
6.3 Cramers Ruleand GeometricInterpretationso
f a Determinant
2
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
3
  • 2 2 determinants and area
  • Recall that the area of the parallelogram spanned
    by a and b is the magnitude of ab. 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

4
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.

5
  • 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 .

6
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)

7
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8
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9
Cramers 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

10
Cramers 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
11
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12
Homework p. 607 (8.5) Pre-Calc book 1-27 odd
13
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