15: Making Moves with Matrices in VS .NET

Creator QR Code 2d barcode in VS .NET 15: Making Moves with Matrices

1 Multiply Row 1 times 3 and add the result to Row 2, making the result the new Row 2 2 Multiply Row 1 by 2 and add the result to Row 3, putting the resulting answer in Row 3 The following shows what the matrix looks like after you perform the row operations The notation on the arrow between the two matrices describes the row operations you use

0 1 0

3 5 6

3 36 0

0 1 0

0 0 1

7 44 2

You see a one in the second column and second row of the resulting matrix along the diagonal of ones that you want Getting the one in this position is a nice coincidence; it doesn t always work out as well If the one hadn t fallen in that position, you d have to divide the whole row by whatever number makes that element a one Now, you want zeros above and below the one middle, so you follow these steps: 1 Multiply Row 2 by 2 and add the result to Row 1 2 Multiply Row 2 by 7 and add the result to Row 3

1 2 0 1

0 1 0

0 0 1

1 0 0

0 1 0

8 5 2 6 3 1 7 2 19

0 0 1

0 7 44 2

You now need to turn the element in the third row, third column of the matrix into a one, so you multiply the row through by 05; this multiplication is the same as dividing through by 2

1 0 0

0 1 0

8 5 2 6 3 1 7 2 19

0 0 1

1 0 0

0 1 0

8 5 6 3

2 1

1 95 35 05

You have one last set of row operations to perform to get zeros above the last one on the diagonal: 1 Multiply Row 3 by 8 and add the elements in the row to Row 1 2 Multiply Row 3 by 6 and add the result to Row 2 You now have the 3 3 identity matrix on the left and a new 3 3 matrix on the right The elements in the matrix on the right form the inverse matrix, M 1 The product of M and its inverse, M 1, is the identity matrix The following figure shows the final two steps and the result

1 0 0

0 1 0

8 5 6 3

2 1

1 0 0

0 1 0

0 81 30 4 0 60 22 3 1 95 35 05

1 95 35 05 4 3

81 30 M =

95 35 05

The following figure checks your work showing that M M 1 is the identity matrix:

81 30 4 22 95 35 05

0 1 0

0 0 1

M * M = 3 5 6 * 60 2 3 36

3 = 0

Here s what happens when a matrix doesn t have an inverse: You can t get ones along the diagonal by using the row operations You usually get a whole row of zeros as a result of your row operations A row of zeros is the warning that no inverse is possible

You have a special rule at your disposal to find inverses of 2 2 matrices To implement the rule for a 2 2 matrix, you have to switch two elements, negate two elements, and divide all the elements by the difference of the cross products of the elements This may sound complicated, but the math is really neat and sweet, and the process is much quicker than the general method (see the previous section)