matrics question

Question

to control a crop deasies it is necessary to use 8 units of chemicals A,14 units of B, 13 unitss of C. One barrel of spray P contains one unit of A,2units of B, 3 units of C, . one barrel of spaary Q contains 2 units of A, 3 units of B, 2 units of C,  one barrel of spary contains 1 unit of A, 2 units of B, 2 units off C. find how many barrels of each spary be used to just met the requirement. (solve by matrics algebra)

 

Answer 1

First, we need to set up the matrix equation.  To do this, we'll create a 3x3 matrix with each of the rows representing a chemical and each column representing a barrel.  We'll multiply this by a 1x3 column that represents the amount of each barrel we need to include.  Then we'll set this equal to our desired composition of chemicals represented by a 1x3 matrix.  So,

 

[ 1  2  1 ]  [ p ]           [  8  ]

[ 2  3  2 ]  [ q ]    =    [  14 ] 

[ 3  2  2 ]  [ r  ]          [  13 ]

 

In order to solve this equation, we need to multiply both sides by the inverse of our 3x3 matrix, which means first we need to find the inverse:

 

[ 1  2  1  | 1  0  0 ]
[ 2  3  2  | 0  1  0 ]
[ 3  2  2  | 0  0  1 ]

 

First we'll subtract 2x the first row from the second and 3x the first row from the third in order to use the 1 in the first column to zero out the rest of its column:

 

[ 1  2  1  | 1  0  0 ]
[ 0 -1  0  |-2  1  0 ]
[ 0 -4 -1 | -3  0 1 ]

 

Then we'll multiply the second row by -1 and use second row to zero out the rest of the second column:

 

[ 1  0  1  | -3  2  0 ]

[ 0  1  0  |  2 -1  0 ]

[ 0  0 -1  |  5 -4  1 ]

 

Then we'll multiply the third row by -1 and then subtract the third row from the first:

 

[ 1  0  0  |  2 -2  1 ]

[ 0  1  0  |  2 -1  0 ]

[ 0  0  1  | -5  4 -1 ]

 

Now, the right half of this is our desired inverse.  If we multiply both sides of our initial equation by this, we get:

 

[p]                [2  -2  1 ]  [ 8  ]                 [ 1 ]

[q]       =       [2  -1  0 ]  [ 14]       =        [ 2 ]

[r]                 [-5  4 -1 ] [ 13 ]                 [ 3 ]

 

If we check back to the original problem, we see that mixing one of barrel P, 2 of barrel Q, and 3 of barrel R gives us the desired mix of chemicals.

 

If that isn't clear or if you have additional questions, please let me know.