FINDING the -INVERSE-OF-MATRIX.pptx

Finding Inverse of Matrix

1. FINDING INVERSE OF MATRIX Prepared by: ANGELICA J. NEGRO

2. O bjectives Verify that multiplying a matrix by its inverse results in 1. Use matrix multiplication to find the inverse of a matrix. Find an inverse by augmenting with an identity matrix.

3. What is an inverse of a matrix? This is the reciprocal of a number.

4. Note: The inverse of A is A -1  only when : A × A -1  = A -1  × A =  I   I t should be a square matrix. Not all square matrices have inverses. A square matrix which has an inverse is called  invertible  or nonsingular, and a square matrix without an inverse is called  noninvertible  or singular.

5. Three ways to Find the Inverse of Matrix STEPS IN SOLVING 2 X 2 MATRIX Switch the numbers in (row 1, column 1) and (row 2, column 2) Give opposite signs to the numbers in (row 1, column 2) and (row 2, column 1) Find the   determinant   in the original matrix Multiply to find the inverse.

6. AUGMENTED METHOD

7. Gauss-Jordan Elimination

8. 3. Adjoint Method Check the determinant of the matrix.  You need to calculate the determinant of the matrix as an initial step. If the determinant is 0, then your work is finished, because the matrix has no inverse .

9. Using Gauss-Jordan Elimination

10. Adjoint Method 2. Transpose the original matrix.  Transposing means reflecting the matrix about the main diagonal, or equivalently, swapping the ( i,j ) th element and the ( j,i ) th. When you transpose the terms of the matrix, you should see that the main diagonal (from upper left to lower right) is unchanged .

11. Using Gauss-Jordan Elimination 3. Find the determinant of each of the 2x2 minor matrices.  Every item of the newly transposed 3x3 matrix is associated with a corresponding 2x2 “minor” matrix. To find the right minor matrix for each term.

12. Create the matrix of cofactors.  Place the results of the previous step into a new matrix of cofactors by aligning each minor matrix determinant with the corresponding position in the original matrix. You must then reverse the sign of alternating terms of this new matrix, following the “checkerboard” pattern shown.When assigning signs, the first element of the first row keeps its original sign. The second element is reversed. The third element keeps its original sign. Continue on with the rest of the matrix in this fashion. Note that the (+) or (-) signs in the checkerboard diagram do not suggest that the final term should be positive or negative. They are indicators of keeping (+) or reversing (-) whatever sign the number originally had. The final result of this step is called the adjugate matrix of the original. This is sometimes referred to as the adjoint matrix. The adjugate matrix is noted as Adj (M).

13. Adjoint Method Divide each term of the adjugate matrix by the determinant.  Recall the determinant of M that you calculated in the first step (to check that the inverse was possible). You now divide every term of the matrix by that value. Place the result of each calculation into the spot of the original term. The result is the inverse of the original matrix.

14. Activity: Find the inverse of the following matrices: 3. A group took a trip on a  bus , at P3 per child and P3.20 per adult for a total of P118.40. They took the  train  back at P3.50 per child and P3.60 per adult for a total of P135.20 . How many children, and how many adults are in a group?