6.3 Cramer’s Rule and Geometric Interpretations of a Determinant.
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Transcript of 6.3 Cramer’s Rule and Geometric Interpretations of a Determinant.
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6.3 Cramer’s Ruleand GeometricInterpretationsof a Determinant
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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 visithttp://www-math.mit.edu/18.013A/HTML/chapter04/section01.html
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• 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
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VolumeDeterminants 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.
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• 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| = .⋅
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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)
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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
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Cramer’s Rule states that
and
Where D is the determinant of ADx is the Determinant of A with the x column replaced by bDy is the Determinant of A with the y column replaced by b
Note: verify that this works by checking with the previous slide
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Homework: p. 607 (8.5) Pre-Calc book 1-27 odd
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