4x + y + z = 6

-x +y - 4z +-19

x + 3y + 4z = 11
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1 Answer

4x + y + z = 6

-x +y - 4z =-19

x + 3y + 4z = 11

 

4x + y + z = 6               | 4  1   1 || x | = |  6|

-x  + y – 4z = -19          |-1  1  -4 || y | = |-19|

 x  + 3y + 4z = 11         | 1   3   4|| z | = | 11|

Starting with MX = R, we find the inverse of M, M^(-1), using which we evaluate the unknowns matrix, X, with the matrix equation, X = M^(-1) * R, where R is the constant matrix, [6 -19 11].

M =      | 4  1  1|            M^T = | 4  -1  1|

            |-1  1 -4|                       | 1   1  3|

            | 1  3  4|                        | 1  -4  4|

Adj(M) = cofactors of M^T.(Or, the transpose of the cofactors of M – same thing)

Adj(M) = |1   3| = 4 + 12              |1   3| = 4 – 3                  |1   1|= -4 – 1

               |-4  4| = 16                    |1   4| =   1                      |1  -4| =  -5

               |-1  1| = -4 + 4               |4   1| = 16 – 1                |4  -1| = -16 + 1

               |-4  4|    0                       |1   4| =  15                    |1  -4| =  -15

               |-1  1| = -3 - 1                |4   1| = 12 - 1                 |4  -1| = 4 + 1

               | 1  3| = -4                      |1   3| = 11                      |1   1| = 5

Adj(M)  = |16    1    -5| x |+  -  +| = |16   -1   -5|

                | 0   15  -15|    |-  +  - |    | 0   15  15|   

                |-4   11     5|    |+  -  +|    |-4  -11    5|   

det(M) = 4|1  -4| - 1|-1  -4| + 1|-1  1| = 4(4 + 12) – 1(-4 +4) + 1(-3 – 1) = 64 + 0 – 4 = 60

                 |3   4|     | 1   4|       | 1  3|

det(M) = 60

Inverse Matrix

M^(-1) = 1/det(M) * Adj(M)

M^(-1) = (1/60) * |16   -1   -5|

                           | 0    15  15|   

                           |-4   -11    5|   

X = M^(-1) * R

X = (1/60)*|16    -1   -5| * |    6| = (1/60) * | 96 +  19  –   55| = (1/60)*|   60| =| 1 |

                  | 0    15  15|    |-19|                 |   0 – 285 + 165|               |-120|   |-2|

                  |-4   -11    5|    |  11|                 |-24 + 209 +  55|               | 240|    | 4|

Solution: x = 1, y = -2, z = 4

 

by Level 11 User (81.5k points)

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