BELL-WORKSolve the system of equations using matrices:

yxxy

10102036114

HW 4.6(d)Due tomorrow:

Handout (even)Calculate the inverse

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)

Guiding question:

How is the inverse of a 3×3 matrix calculated?

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.

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.

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.

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

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

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

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 =

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

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

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

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 =

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

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

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

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 =

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

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…

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

The Matrix of CofactorsIn the Matrix of Minors, change the sign of alternate cells as follows:

The Matrix of CofactorsIn the Matrix of Minors, change the sign of alternate cells as follows:

The Matrix of CofactorsMatrix of minors =

0100332

222

The Matrix of CofactorsMatrix of cofactors =

0100332222

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

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

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

The Determinant of a 3×3 Matrixis given by

The Determinant of a 3×3 Matrixis given by

The Determinant of a 3×3 Matrixis given by

– +

3×2

The Determinant of a 3×3 Matrixis given by

– +

3×2 – 0×2

The Determinant of a 3×3 Matrixis given by

– +

3×2 – 0×2 + 2×2

The Determinant of a 3×3 Matrixis given by

– +

3×2 – 0×2 + 2×2 = 10

The Inverseis given by

0321032

022

101

The Inverseis given by

01

0

103

51

103

51

51

51

The InverseHandout (odd)

Who wants to answer the Guiding question?

How is the inverse of a 3×3 matrix calculated?