BELL-WORK

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BELL-WORK

• Solve the system of equations using matrices

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HW 4.6(d)

• Due tomorrow
• Handout (even)
• Calculate the inverse

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HW 4.6(c) Solutions

• 5. (2,-1)
• 7. (0,-5)
• 9. (3,-5)
• (06,-7)
• (3,8)
• (2,-1)
• (6,-6)
• (1,1)
• (-1,-1)

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Guiding question

• How is the inverse of a 33 matrix calculated?

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Matrices

• We can use matrices to solve a system of 3 linear
  equations!
• Therefore, we need to be able to find the inverse
  of a 33 matrix.
• This is a more complicated process than for a 22
  matrix.

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Finding the Inverse of a 33 Matrix

• Step 1 Calculating the Matrix of Minors.
• Step 2 Turn that into the Matrix of Cofactors.
• Step 3 Turn that matrix into the Adjugate.
• Step 4 Multiply that by 1/Determinant of the
  original matrix.

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The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.

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The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for

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The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for

10
The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for

11
The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for
• Determinant

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The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for
• Determinant 2

13
The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for

14
The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for

15
The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for
• Determinant

16
The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for
• Determinant 2

17
The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for

18
The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for

19
The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for
• Determinant

20
The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Example Calculate the matrix of minors for
• Determinant 2

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The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Continue this process

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The Matrix of Minors

• For each element of the matrix, the matrix of
  minors is formed by ignoring the values on the
  current row and column and calculating the
  determinant of the remaining values.
• Matrix of minors

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The Matrix of Cofactors

• In the Matrix of Minors, change the sign of
  alternate cells as follows

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The Matrix of Cofactors

• In the Matrix of Minors, change the sign of
  alternate cells as follows

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The Matrix of Cofactors

• Matrix of minors

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The Matrix of Cofactors

• Matrix of cofactors

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The Adjugate

• Transpose all elements of the matrix of cofactors
  by swapping their positions over the diagonal
  (the diagonal stays the same).

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The Adjugate

• Transpose all elements of the matrix of cofactors
  by swapping their positions over the diagonal
  (the diagonal stays the same).
• Matrix of cofactors

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The Adjugate

• Transpose all elements of the matrix of cofactors
  by swapping their positions over the diagonal
  (the diagonal stays the same).
• Matrix of cofactors
• Adjugate

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The Determinant of a 33 Matrix

• is given by

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The Determinant of a 33 Matrix

• is given by

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The Determinant of a 33 Matrix

• is given by
• 32

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The Determinant of a 33 Matrix

• is given by
• 32 02

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The Determinant of a 33 Matrix

• is given by
• 32 02 22
  

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The Determinant of a 33 Matrix

• is given by
• 32 02 22
  10

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The Inverse

• is given by

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The Inverse

• is given by

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The Inverse

• Handout (odd)

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Who wants to answer the Guiding question?

• How is the inverse of a 33 matrix calculated?