BELL-WORK Solve the system of equations using matrices:
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Transcript of BELL-WORK Solve the system of equations using matrices:
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BELL-WORKSolve the system of equations using matrices:
yxxy
10102036114
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HW 4.6(d)Due tomorrow:
Handout (even)Calculate the inverse
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HW 4.6(c) Solutions5. (2,-1)7. (0,-5)9. (3,-5)12. (06,-7)14. (3,8)16. (2,-1)18. (6,-6)20. (1,1)22. (-1,-1)
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Guiding question:
How is the inverse of a 3×3 matrix calculated?
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MatricesWe can use matrices to solve a system of 3 linear
equations!
Therefore, we need to be able to find the inverse of a 3×3 matrix.
This is a more complicated process than for a 2×2 matrix.
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Finding the Inverse of a 3×3 MatrixStep 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 MinorsFor 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 MinorsFor 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
110202
203
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The Matrix of MinorsFor 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
![Page 10: BELL-WORK Solve the system of equations using matrices:](https://reader035.fdocuments.in/reader035/viewer/2022062504/5a4d1b4b7f8b9ab0599a55c8/html5/thumbnails/10.jpg)
The Matrix of MinorsFor 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
![Page 11: BELL-WORK Solve the system of equations using matrices:](https://reader035.fdocuments.in/reader035/viewer/2022062504/5a4d1b4b7f8b9ab0599a55c8/html5/thumbnails/11.jpg)
The Matrix of MinorsFor 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 =
![Page 12: BELL-WORK Solve the system of equations using matrices:](https://reader035.fdocuments.in/reader035/viewer/2022062504/5a4d1b4b7f8b9ab0599a55c8/html5/thumbnails/12.jpg)
The Matrix of MinorsFor 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 MinorsFor 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
![Page 14: BELL-WORK Solve the system of equations using matrices:](https://reader035.fdocuments.in/reader035/viewer/2022062504/5a4d1b4b7f8b9ab0599a55c8/html5/thumbnails/14.jpg)
The Matrix of MinorsFor 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
![Page 15: BELL-WORK Solve the system of equations using matrices:](https://reader035.fdocuments.in/reader035/viewer/2022062504/5a4d1b4b7f8b9ab0599a55c8/html5/thumbnails/15.jpg)
The Matrix of MinorsFor 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 =
![Page 16: BELL-WORK Solve the system of equations using matrices:](https://reader035.fdocuments.in/reader035/viewer/2022062504/5a4d1b4b7f8b9ab0599a55c8/html5/thumbnails/16.jpg)
The Matrix of MinorsFor 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
![Page 17: BELL-WORK Solve the system of equations using matrices:](https://reader035.fdocuments.in/reader035/viewer/2022062504/5a4d1b4b7f8b9ab0599a55c8/html5/thumbnails/17.jpg)
The Matrix of MinorsFor 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
![Page 18: BELL-WORK Solve the system of equations using matrices:](https://reader035.fdocuments.in/reader035/viewer/2022062504/5a4d1b4b7f8b9ab0599a55c8/html5/thumbnails/18.jpg)
The Matrix of MinorsFor 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
![Page 19: BELL-WORK Solve the system of equations using matrices:](https://reader035.fdocuments.in/reader035/viewer/2022062504/5a4d1b4b7f8b9ab0599a55c8/html5/thumbnails/19.jpg)
The Matrix of MinorsFor 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 =
![Page 20: BELL-WORK Solve the system of equations using matrices:](https://reader035.fdocuments.in/reader035/viewer/2022062504/5a4d1b4b7f8b9ab0599a55c8/html5/thumbnails/20.jpg)
The Matrix of MinorsFor 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
![Page 21: BELL-WORK Solve the system of equations using matrices:](https://reader035.fdocuments.in/reader035/viewer/2022062504/5a4d1b4b7f8b9ab0599a55c8/html5/thumbnails/21.jpg)
The Matrix of MinorsFor 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 MinorsFor 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 =
0100332
222
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The Matrix of CofactorsIn the Matrix of Minors, change the sign of alternate cells as follows:
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The Matrix of CofactorsIn the Matrix of Minors, change the sign of alternate cells as follows:
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The Matrix of CofactorsMatrix of minors =
0100332
222
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The Matrix of CofactorsMatrix of cofactors =
0100332222
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The AdjugateTranspose all elements of the matrix of cofactors by swapping their positions over the diagonal (the diagonal stays the same).
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The AdjugateTranspose all elements of the matrix of cofactors by swapping their positions over the diagonal (the diagonal stays the same). Matrix of cofactors =
0100332222
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The AdjugateTranspose all elements of the matrix of cofactors by swapping their positions over the diagonal (the diagonal stays the same). Matrix of cofactors =
Adjugate =
0100332222
0321032
022
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The Determinant of a 3×3 Matrixis given by
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The Determinant of a 3×3 Matrixis given by
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The Determinant of a 3×3 Matrixis given by
– +
3×2
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The Determinant of a 3×3 Matrixis given by
– +
3×2 – 0×2
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The Determinant of a 3×3 Matrixis given by
– +
3×2 – 0×2 + 2×2
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The Determinant of a 3×3 Matrixis given by
– +
3×2 – 0×2 + 2×2 = 10
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The Inverseis given by
0321032
022
101
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The Inverseis given by
01
0
103
51
103
51
51
51
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The InverseHandout (odd)
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Who wants to answer the Guiding question?
How is the inverse of a 3×3 matrix calculated?