Cramer

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• Cramers Rule for 2x2 systems

Jerry, I'm at the corner of 1st and 1st. Wait a
minute, how can a street intersect itself? I
must be at the nexus of the universe!
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Cramers Rule - 2 x 2

• Cramers Rule relies on determinants.
• Consider the system below with variables x and y

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Cramers Rule - 2 x 2

• We can change this system of equations into
  matrix multiplication

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Cramers Rule - 2 x 2

• We can solve for x in the system below by finding
  determinants.
• The first is the determinant of the coefficient
  matrix we already know how to find this

Original matrix multiplication
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Cramers Rule - 2 x 2

• Next, we need to find the determinant of a
  special matrix for x. We get this matrix by
  substituting the x values from the coefficient
  matrix out and the answer values in

Original matrix multiplication
New matrix with x values gone and answers
instead
Answers subbed in
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Cramers Rule - 2 x 2

• And we can find the determinant of this new,
  x-swapped matrix

Original matrix multiplication
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Cramers Rule - 2 x 2

• Now we can solve for x by using the determinant
  of the original (coefficient) matrix and the
  determinant of the new, x-swapped matrix. This
  is done according to this formula

Original matrix multiplication
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Cramers Rule - 2 x 2

• Example solve for x in the following system

We know the original matrix, we just need to find
the x-swapped matrix
And we are ready to solve for x by using
determinants
x 2
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Cramers Rule - 2 x 2

• The same holds true for y as well as x. We just
  need to swap the answers in for the y-values
  instead. Therefore, Cramers Rule is summarized
  as follows

Given the following matrix multiplication
As long as the determinant of the coefficient
(original) matrix does not equal zero
Then
and
If it does equal zero, you cannot use Cramers
Rule to solve the system.