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Name: Jamie
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Question:
1) how do you find the inverse of a 3x3 matrix? 2) what is the definition of a 3x3 triangular matrix (in english, please)?



Replies:
1)  An nxn matrix is invertible if it is row-equivalent to the nxn identity
matrix.  This means if you can transform a matrix into the identity matrix
by performing row operations, it is invertible.
Row operations include:
   1.  Multiplying a row by a constant
   2.  Swapping two rows
   3.  Multiplying a row by a constant and adding the result to another row.

Generally the inverse is found by writing the identity matrix beside the
matrix you want to find the inverse of, and then attempting to transform the
matrix into the identity matrix, while performing the same operations on the
identity matrix.  As long as you don't get a row of all zeroes, you will be
able to calculate the inverse.  When you are done, the original matrix
should be the identity matrix, and the original identity matrix will be the
inverse.

For example:

Given this matrix:
         2 9 6
         1 4 3
         2 2 0

First write it and the identity matrix together:

         2 9 6 | 1 0 0
         1 4 3 | 0 1 0
         2 2 0 | 0 0 1

Then:
   1. Swap row 1 and 2
         1 4 3 | 0 1 0
         2 9 6 | 1 0 0
         2 2 0 | 0 0 1

   2. Multiply row 1 by -2 and add it to row two
         1 4 3 | 0 1 0
         0 1 0 | 1 -2 0
         2 2 0 | 0 0 1

   3. Multiply row 1 by -2 and add it to row three
         1 4 3 | 0 1 0
         0 1 0 | 1 -2 0
         0 -6 -6 | 0 -2 1

   4. Multiply row 3 by -0.5 and add it to row 1
         1 1 0 | 0 0 0.5
         0 1 0 | 1 -2 0
         0 -6 -6 | 0 -2 1

   5. Multiply row 2 by -1 and add it to row 1
         1 0 0 | -1 2 0.5
         0 1 0 | 1 -2 0
         0 -6 -6 | 0 -2 1

   6. Multiply row 2 by 6 and add it to row 3
         1 0 0 | -1 2 0.5
         0 1 0 | 1 -2  0
         0 0 -6 | 6 -14 1

   7. Multiply row 3 by -1/6
         1 0 0 | -1 2  1/2
         0 1 0 | 1 -2   0
         0 0 1 | -1 14/6 -1/6


So, the inverse of
         2 9 6
         1 4 3
         2 2 0

is:
  -1 2  1/2
   1 -2   0
  -1 14/6 -1/6


You can verify this by doing the matrix multiplication
         2 9 6      -1 2  1/2             1 0 0
         1 4 3  x   1 -2   0          =  0 1 0
         2 2 0      -1 14/6 -1/6               0 0 1


Also, this process will work for any square matrix, not just one that is
3x3.


2)
An upper triangular matrix is a square matrix having only zeros below its
main diagonal.  A lower triangular matrix is a square matrix having only
zeroes above its main diagonal.  A triangular matrix is either upper or
lower triangular.

For example:

   2 3 5
   0 1 4   Is an upper triangular matrix.  All the non-zero numbers are in
the upper right half.
   0 0 6

Eric Tolman
Computer Scientist




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