X+2y-2z=-11 2x +z =-6 5y -3z = -7 using a matrix: 1 2 -2| -11 2 0 1 | -7 0 5 -3 | -6
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X+2y-2z=-11 2x +z =-6 5y -3z = -7 using a matrix: 1 2 -2| -11 2 0 1 | -7 0 5 -3 | -6

 

x + 2y – 2z = -11          | 1  2  -2|| x | = | -11|

2x        +  z = -6            | 2  0   1 || y | = | -6|

      5y – 3z = -7            | 0   5  -3|| z | = | -7|

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, [-11 -6 -7].

M =      | 1  2  -2|          M^T = | 1  2   0|

            | 2  0   1|                      | 2  0   5|

            | 0  5  -3|                      |-2  1  -3|

Adj(M) = |0  5| = 0 – 5   | 2    5| = -6 + 10             | 2   0| = 2 – 0

               |1 -3| = -5        |-2  -3| =   4                     |-2  1| =  2

               |2  0| = -6 – 0  | 1    0| = -3 – 0                | 1  2| = 1 + 4

               |1 -3|    -6        |-2  -3| =  -3                     |-2  1| =  5

               |2  0| = 10        | 1    0| = 5                       | 1  2| = -4

               |0  5|                | 2    5|                              | 2  0|

Adj(M)  = |-5   4   2| x |+  -  +| = |-5   -4   2|

                |-6  -3  5|     |-  +  -|     | 6   -3  -5|   

                |10  5 -4|     |+  -  +|    |10  -5 -4|   

det(M) = 1|0   1| - 2|2   1| - 2|2  0| = (0 – 5) – 2(-6 – 0) – 2(10 – 0) = -5 + 12 – 20 = -13

                 |5 -3|      |0  -3|     |0  5|

det(M) = -13

Inverse Matrix

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

M^(-1) = (-1/13) * |-5   -4   2|

                             | 6   -3  -5|   

                             |10  -5 -4|   

X = M^(-1) * R

X = (-1/13)*|-5   -4   2| * | -11 | = (-1/13) * |  55 + 24 – 14| = (-1/13)*| 65| = |-5 |

                   | 6   -3  -5|    |   -6 |                 |-66 + 18 + 35|                 |-13|   |  1 |

                   |10  -5 -4|     |   -7 |                 |-110 + 30 + 28|                |-52|   |  4 |

Solution: x = -5, y = 1, z = 4

 

by Level 11 User (81.5k points)

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